EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO BOB OLIVER Abstract. We prove here the Martino-Priddy conjecture for the prime 2: th* *e 2- completions of the classifying spaces of two groups G and G0are homotopy * *equivalent if and only if there is an isomorphism between their Sylow 2-subgroups wh* *ich preserves fusion. This is a consequence of a technical algebraic result, which says* * that for a finite group G, the second higher derived functor of the inverse limit va* *nishes for a certain functor ZG on the 2-subgroup orbit category of G. The proof of th* *is result uses the classification theorem for finite simple groups. In their paper [MP ], John Martino and Stewart Priddy conjectured that for a* *ny prime p and any pair G, G0 of finite groups, the p-completed classifying spaces* * BG^p and BG0^pare homotopy equivalent if and only if the p-local structures of G and* * of G0 are isomorphic in a sense to be made precise below. In an earlier paper [BLO* * ] in collaboration with Carles Broto and Ran Levi, we identified the obstruction gro* *ups to constructing a homotopy equivalence between these spaces, given an isomorphi* *sm between the p-local structures. For odd primes p, these groups have already be* *en shown [Ol] to vanish in all cases. The main technical result of this paper is t* *hat these obstruction groups also vanish when p = 2. The proof of this result (like the p* *roof of the conjecture for odd primes) depends on the classification theorem for finite* * simple groups. Fix a prime p and a finite group G. For any pair of subgroups P, Q G, let N* *G(P, Q) denote the transporter: NG(P, Q) = {x 2 G | xP x-1 Q}. The p-subgroup orbit category of G is the category Op(G) whose objects are the* * p- subgroups of G, and where MorOp(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q). A p-subgroup P G is called p-centric if Z(P ) is a Sylow p-subgroup of CG(P )* *, or equivalently if CG(P ) = Z(P ) x C0G(P ) for some subgroup C0G(P ) of order pri* *me to p. Define the functor ZG : Op(G) op------! Ab by setting ZG(P ) = Z(P ) if P is p-centric in G and ZG(P ) = 0 otherwise, and * *letting c-1x cx 2 Mor Op(G)(P, Q) induce the morphism Z(Q) ---! Z(P ) if P and Q are both p- centric. Our main algebraic result is the following. Theorem A. For any finite group G, lim-i(ZG) = 0 for all i 2. O2(G) ___________ 1991 Mathematics Subject Classification. Primary 55R37. Secondary 55P60, 20D0* *5. Key words and phrases. Classifying space, p-completion, finite simple groups. Partially supported by UMR 7539 of the CNRS. 1 2 BOB OLIVER Proof.In Proposition 2.9, we show that lim-i(ZG) = 0 for all i 2 if each nona* *belian simple group L which appears in the decomposition series of G belongs to a cert* *ain class L 2(2). We then show that L 2(2) contains all alternating groups (Theorem* * 5.1); all simple groups of Lie type in characteristic two including the Tits group (T* *heorem 6.3); all simple groups of Lie type in odd characteristic (Theorems 7.4 and 8.1* *1); and all sporadic groups (Theorem 9.1). Theorem A then follows from the classificat* *ion theorem for finite simple groups. Theorem A was motivated by studying equivalences between completed classifying spaces of finite groups. Let p be a prime, let G and G0be finite groups, and le* *t S G ~= and S0 G0 be Sylow p-subgroups. An isomorphism ': S --! S0 is called fusion preserving if for all P, Q S and all P -ff-!~Q, ff is conjugation by an elem* *ent of G = 'ff'-1 0 if and only if '(P ) ---!~ '(Q) is conjugation by an element of G . = The Martino-Priddy conjecture states that for any prime p, and any pair G, G0* * of finite groups, BG^p' BG0^pif and only if there is a fusion preserving isomorphi* *sm between Sylow p-subgroups of G and G0. The ö nly if" part of the conjecture was proved by Martino and Priddy [MP ], and follows from the bijection ~= ^ Rep (P, G) def=Hom(P, G)= Inn(G) ------! [BP, BGp] for any p-group P and any finite group G [BL , Proposition 2.1]. Conversely, by* * [BLO , Proposition 6.1], given a fusion preserving isomorphism between Sylow p-subgrou* *ps of G and G0, the obstruction to extending it to a homotopy equivalence BG^p' BG0^p* *lies in lim-2(ZG). Hence Theorem A implies: Theorem B (Martino-Priddy conjecture at the prime 2). For any pair G and G0 of finite groups with Sylow 2-subgroups S G and S0 G0, BG^2' BG0^2if and only * *if ~= there is a fusion preserving isomorphism S --! S0. We next turn to the question of self equivalences of BG^p. For any space X, * *let Out(X) denote the group of homotopy classes of self homotopy equivalences of X. For any finite group G, any prime p, and any Sylow p-subgroup S G, let Autfus* *(S) be the group of fusion preserving automorphisms of S, let Aut G(S) be the group* * of automorphisms induced by conjugation by elements of G (i.e., elements of NG(S))* *, and set Outfus(S) = Autfus(S)= AutG(S). Theorem A, when combined with [BLO , Theorem 6.2], gives the following descrip* *tion, up to extension, of Out(BG^p). Theorem C. For any finite group G with Sylow 2-subgroup S G, there is a short exact sequence 1 ---! lim-1(ZG) -----! Out (BG^2) -----! Out fus(S) ---! 1. O2(G) In [Ol], we showed that when p is odd, the groups lim-i(ZG) vanish for all fi* *nite G and all i 1. In contrast, when p = 2, the groups lim-1(ZG) can be nonvanishi* *ng. Examples of this include the groups G = P SL2(q) when q 1 (mod 8), G = An when n 2, 3 (mod 4), and G = P SL4(q) when q 3 (mod 4). A simple proof of t* *his EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 3 when G has dihedral Sylow 2-subgroup is given in Proposition 1.6, and other cas* *es are discussed in Section 10. If G is an arbitrary finite group, then ZG can be filtered by subfunctors ZKG* *, defined for all K C G by setting ( Z(P ) \ K if P is p-centric in G ZKG(P ) = 0 otherwise. The idea of the proof of Theorem A is to filter G by a maximal sequence 1 = K0 K1 . . .Kn-1 Kn = G of normal subgroups, and then analyze higher limits of the ZKjG=ZKj-1G. In particular, lim-i(ZG) = 0 for all i 2 if the same holds f* *or higher limits of ZKjG=ZKj-1Gfor all j. Kj Kj-1 We first reduce the general computation of lim-*ZG =ZG to the special case where L = Kj is quasisimple (i.e., L is perfect and L=Z(L) is simple) with p-gr* *oup center A = Kj-1 = Z(L). This is done in Lemmas 2.1 and 2.4. We then observe that in this case, lim-*ZLG=ZAG depends only on L and on AutG (L). This motivat* *es the definition of new functors YL on Op( ), defined for any quasisimple group L* * with p-group center and any Aut(L) which contains Inn(L), with the property that lim-*(ZLG=ZAG) ~=lim-*(YL) for any A C L C G as above with = Aut G(L). For example, if L is simple (and identified with Inn(L) C ), then YL = ZL . The definition of YL in the general* * case is given in Definition 2.5. To simplify notation in the rest of the paper, we then define Li(p) to be the* * class of simple groups L for which lim-i(YeL) = 0 for all choices of central extensions * *eLof L and Aut(eL). We are thus reduced to proving that all simple groups lie in L 2(2* *). In Section 4, we define, for each nonabelian finite simple group L, certain s* *ets Ri(L ; p) of p-subgroups which could öc ntribute" to lim-i(-), in a way made pr* *ecise in Definition 4.1. We then show (Proposition 4.2) that L 2 Li(p) if there is a * *p-centric subgroup Q L, which is weakly closed in a Sylow p-subgroup which contains it,* * and with the property that all subgroups in Ri(L ; p) contain Q up to conjugacy. Th* *is is the result which in almost all cases will be used to show L 2 L 2(2) (simple gr* *oups of Lie type in characteristic two are handled in a different way). The last half o* *f Section 4 then consists of a series of propositions, each of which gives some condition* *s to be used when proving that certain subgroups do not lie in R 2(L ; 2). In this way, the paper splits into two halves. Sections 1-4 involve homologi* *cal algebra, and reduce the problem to a series of criteria stated in purely group * *theoretic terms. These criteria are then applied to the individual groups in Sections 5-* *9, to show that L 2 L 2(2) in all cases. Afterwards, in Section 10, some computations* * of lim-1(ZG) are listed (mostly without proof). Clearly, as a topologist writing a paper which depends very heavily on the st* *ructure of the individual finite simple groups, I had a lot of assistance. Michael Asch* *bacher, Ron Solomon, and Richard Lyons all gave extensive help in answering my questions about the structure of certain simple groups. I am grateful to Sergey Shpektoro* *v and Ulrich Meierfrankenfeld for sending me their manuscript listing the maximal 2-l* *ocal subgroups of the monster and the baby monster _ which allowed me to fill in the last step in the proof of Theorem A. I had several helpful discussions with Ge* *orge 4 BOB OLIVER Glauberman and Jesper Grodal during my short visits to the University of Chicag* *o. I am especially indebted to my earlier collaborator, Yoav Segev, who (while not i* *nvolved in this project) taught me much of what I know about the classification theorem* *, and especially about the finite simple groups of Lie type. Finally, I would like t* *o thank my colleagues at Northwestern University and the University of Wisconsin for th* *eir hospitality while working on many of the later stages of this project. General notation: We list, for easy reference, the following notation which w* *ill be used throughout the paper. o Sp(G) denotes the set of p-subgroups of G o Sylp(G) denotes the set of Sylow p-subgroups of G o Gp denotes a Sylow p-subgroup of the group G, but only when it is abelian and* * a direct factor of G o Op(G) is the maximal normal p-subgroup of G o Cn, Dn, and Qn denote cyclic, dihedral, and quaternion groups of order n o An and n are the alternating and symmetric groups on n elements o A radical p-subgroup of G is a p-subgroup P G such that Op(NG(P )=P ) = 1 n o n(P ) (for a p-group P ) is the subgroup generated by all g 2 P such that gp* * = 1 o NG(H, K) = {x 2 G | xHx-1 K} (for H, K G) o cx denotes conjugation by x (g 7! xgx-1) fi o Hom G (H, K) = cx 2 Hom (H, K) fix 2 NG(H, K) (for H, K G) o Aut G(H) = Hom G(H, H) ~=NG(H)=CG(H), and OutG(H) = AutG (H)= Inn(H) o A functor F :Cop ___! Ab is called acyclic if lim-i(F ) = 0 for all i > 0. In the later sections, we also use the following standard shorthand notation * *for referring to certain groups: o 2n = Cn2is an elementary abelian 2-group; o 21+2k+is the central product of k copies of D8; o 21+2k-is the central product of Q8 with k- 1 copies of D8; o 2a+b is a 2-group P such that Z(P ) ~=2a and P=Z(P ) ~=2b; o [2n] is an unspecified group of order 2n; and o H:K, H.K, and H.K are extensions (split, unsplit, or indeterminate, respectiv* *ely) with kernel H and quotient K. Contents 1. Higher limits over orbit categories * * 5 2. Reduction to simple groups 12 3. A relative version of -functors * *19 4. Subgroups which contribute to higher limits * *25 EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 5 5. Alternating groups 33 6. Groups of Lie type in characteristic two * *34 7. Classical groups of Lie type in odd characteristic * * 41 8. Exceptional groups of Lie type in odd characteristic * * 46 9. Sporadic groups 58 10. Computations of lim1(ZG) 69 References 70 1. Higher limits over orbit categories We first fix our notation. For any prime p and any finite group G, the p-subg* *roup orbit category of G is the category Op(G) whose objects are the p-subgroups of * *G, and where MorOp(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q). Recall that NG(P, Q) = {x 2 G | xP x-1 Q} (the transporter). This can also be thought of as a category whose objects are orbits G=P of G and whose morphisms * *are G-maps, but for our purposes it is more convenient to let the objects be subgro* *ups. j 0 j# 0 For any homomorphism G ---! G of groups, we let Op(G) ----! Op(G )denote the induced functor between orbit categories. This section contains a variety of different results whose main point in comm* *on is that they all involve higher limits of functors over orbit categories. 1.1___The_functor__*_ In [JMO ], certain graded Z(p)-modules *(G; M) are def* *ined, for any prime p, any finite group G, and any Z(p)[G]-module M, by setting ( M if P = 1 *(G; M) = lim-*( GM) where GM(P ) = Op(G) 0 otherwise. Note that these depend on the prime p, even though that has been suppressed from the notation. We first list some of the basic properties of these groups. Proposition 1.1. The following hold for any prime p and any finite group G. (a)Fix a p-subgroup P G, and let F : Op(G) ___! Z(p)-mod be any functor which vanishes except on subgroups conjugate to P . Then lim-*(F ) ~= *(NG(P )=P ; F (P )). Op(G) (b)If H C G is a normal subgroup which acts trivially on the Z(p)[G]-module M, * *then ( *(G=H; M) if (p, |H|) = 1 *(G; M) ~= 0 otherwise. (c)If Op(G) 6= 1 (if G contains a nontrivial normal p-subgroup), then *(G; M) * *= 0 for all Z(p)[G]-modules M. 6 BOB OLIVER (d)A short exact sequence 0 --! M0 ---! M ---! M00--! 0 of Z(p)[G]-modules in- duces a long exact sequence . . .---! i(G; M0) ---! i(G; M) ---! i(G; M00) ---! i+1(G; M0) ---! . ... Proof.See [JMO , Propositions 5.4, 5.5, & 6.1]. Proposition 1.1 is usually applied by filtering an arbitrary functor on the o* *rbit cat- egory Op(G) in such a way that all quotient functors vanish except on one conju* *gacy class. By Proposition 1.1(a), the higher limits of these quotient functors can * *then be described in terms of the graded groups *. In most cases, this will be applied* * via the following lemma. Lemma 1.2. The following hold for any prime p, any finite group G, and any func* *tors F, F 0: Op(G) op___! Z(p)-mod. (a)Let ': F ___! F 0be a natural morphism of functors such that for all P 2 Sp* *(G), *(NG(P )=P ; Ker('(P ))) = 0 and *(NG(P )=P ; Coker('(P ))) = 0. Then ' induces an isomorphism lim-*(F ) ~=lim-*(F 0). (b)Let C Op(G) be a full subcategory with the property that Mor Op(G)(P, Q) * *= ? whenever P 2 Ob (C) and Q =2Ob (C). Assume *(NG(P )=P ; F (P )) = 0 for all* * P not in C. Then lim-*(F ) ~=lim-*(F |C). (1) Op(G) C Proof.In the situation of (a), lim-*(Ker (')) = 0 and lim-*(Coker(')) = 0 by Pr* *oposition 1.1(a), together with the obvious filtration of these functors and the exact se* *quences for higher limits of extensions of functors. So lim-*(F ) ~=lim-*(Im (')) ~=lim* *-*(F 0). Now assume C Op(G) and F are as described in (b), and let F 0be the quotient functor F 0(P ) = F (P ) if P 2 Ob (C) and F 0(P ) = 0 otherwise. By (a), the p* *rojection F - i F 0induces an isomorphism lim-*(F ) ~=lim-*(F 0). Also, any injective res* *olution of F 0|C can be extended to an injective resolution of F 0by assigning all func* *tors the value zero on objects not in C, and this shows that lim-*(F 0) ~=lim-*(F |C). Proposition 1.1(b) motivates the following definition. Definition 1.3. For any finite group G and any Z(p)[G]-module M, we say that G * *acts p-faithfully on M if the kernel of the action Ker[G ___! Aut(M)] has order pri* *me to p. The following lemma is an immediate consequence of Proposition 1.1(b,d). Lemma 1.4. If the action of a finite group G on a finite Z(p)[G]-module M is not p-faithful, then *(G; M) = 0. There is also a connection between p-faithful actions and p-centric subgroups. Lemma 1.5. Fix a finite group G and a p-subgroup P G. If NG(P )=P acts p- faithfully on Z(P ), then P is p-centric in G. More generally, if H C G is a no* *rmal subgroup and NG(P )=P acts p-faithfully on Z(P ) \ H, then P \ H is p-centric i* *n H. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 7 Proof.We prove the second statement; the first then follows in the case H = G. * *Set K = CH (P \ H). If P \ H is not p-centric in H, then Z(P \ H) = K \ P is not a Sylow p-subgroup of K, and hence NKP (P )=P has order a multiple of p by Lemma 1.11. But NKP (P )=P acts trivially on Z(P ) \ H, and this contradicts the assu* *mption that N(P )=P acts p-faithfully on Z(P ) \ H. This shows that P \ H is p-centri* *c in H. To give a quick illustration of how these techniques are applied, we describe* * the computation of lim-1(ZG) in certain very simple cases. In particular, the foll* *owing proposition shows that these groups are nonvanishing when G is any of the simple groups A6, A7, or P SL2(q) for q 1 (mod 8). Proposition 1.6. Fix a finite group G and a Sylow subgroup S 2 Syl2(G). Assume * *that S is a dihedral group of order 8, and let T1, T2 be S-conjugacy class represe* *ntatives for the subgroups isomorphic to C22. Then lim-1(ZG) ~=Z=2 if AutG (Ti) = Aut(Ti) for i = 1, 2, and lim-1(ZG) = 0 otherwise. Proof.Assume P S is such that *(NG(P )=P ; ZG(P )) 6= 0. Then P is 2-centric* * in G, so P CG(P )=P has odd order, and *(Out G(P ); ZG(P )) ~= *(NG(P )=P CG(P ); Z(P )) ~= *(NG(P )=P ; Z(P )) 6=* * 0 by Proposition 1.1(b). Hence O2(Out G(P )) = 1 (Proposition 1.1(c)). If P is cy* *clic of order 4 or dihedral of order 8, then Out(P ) is a nontrivial 2-group, so th* *ese cases cannot occur. So we are left only with the cases 8 >: 0 otherwise. The computation 1( 3; (Z=2)2) ~=Z=2 follows from [JMO , Proposition 6.2(i)]. By Alperin's fusion theorem (or the theorem of Burnside shown in [Gor , Theor* *em 7.1.1]), T1 and T2 cannot be G-conjugate. So using Proposition 1.1(a) and the o* *bvious filtration of ZG by subgroups, we obtain an exact sequence 0 ---! lim-0(ZG) ----! Z=2 ----! (Z=2)k ----! lim-1(ZG) ---! 0, O2(G) O2(G) where k is the number of i = 1, 2 such that Aut G(Ti) ~= 3. If k 1, then Z(* *S) is G-conjugate to another subgroup of S, and this implies that lim-0(ZG) = 0. * *Thus lim-1(ZG) has rank k- 1 in this case, and is trivial otherwise. 1.2___Reduction_to_smaller_orbit_categories_ We next list some conditions which allow us, in certain situations, to reduce the computation of higher limits of * *a functor on Op(G) to those of another functor on Op(N(Q)=Q) for some Q G. Lemma 1.7. Fix a finite group G and a p-subgroup Q G. Then there is a well defined functor GQ: Op(NG(Q)=Q) ------! Op(G) 8 BOB OLIVER such that GQ(P=Q) = P for all P=Q NG(Q)=Q. Let T be any set of p-subgroups * *of G with the property P 2 T =) Q C P , and Q C xP x-1 for x 2 G implies x 2 NG(Q). (*) Then for any functor F : Op(G) op___! Z(p)-mod which vanishes except on subgro* *ups G-conjugate to elements of T , the induced homomorphism GQ* * G lim-*(F ) ------!~ lim- (F O Q) (1) Op(G) = Op(NG(Q)=Q) is an isomorphism. Proof.Set = GQfor short. Clearly, is well defined on objects. To see that * *it is well defined on morphisms, recall first that Mor Op(G)(P, P 0) = P 0\NG(P, P 0), where NG(P, P 0) is the set of all x 2 G such that xP x-1 P 0. Hence for any * *pair of objects P=Q and P 0=Q in Op(NG(Q)=Q) , Mor Op(NG(Q)=Q)(P=Q, P 0=Q) = (P 0=Q)\NN(Q)=Q(P=Q, P 0=Q) ~=P 0\NN(Q)(P, P 0) P 0\NG(P, P 0) = Mor Op(G)(P, P 0); and is defined on morphism sets to be this inclusion. Composition with is natural in F and preserves short exact sequences of fun* *ctors. Hence if F 0 F is a pair of functors from Op(G) to Z(p)-mod, and the lemma hol* *ds for F 0and for F=F 0, then it also holds for F by the 5-lemma. It thus suffices* * to prove that (1) is an isomorphism when F vanishes except on the G-conjugacy class of o* *ne subgroup P 2 T . When P = Q, then (1) is precisely the isomorphism lim-*(F ) ~= *(N(Q)=Q; F (Q)) of Proposition 1.1(a). Now let P 2 T be arbitrary. By condition (*), Q C P , NG(P ) NG(Q), and F O vanishes except on the Op(NG(Q)=Q) -isomorphism class of P=Q. Let 0= N(Q)=QP=Q: Op(NG(P )=P )------! Op(NG(Q)=Q) be the functor 0(R=P ) = R=Q for p-subgroups R NG(P ) NG(Q) containing P . Then the following square commutes * * lim-*(F_)______! lim- (F O ) Op(G) Op(N(Q)=Q) ( O 0)*~=|| 0*~=|| # # *(NG(P )=P ; F (P=))= *(NG(P )=P ; F (P )) , and the vertical maps are isomorphisms by Proposition 1.1(a) (see the proof of * *[JMO , Lemma 5.4] for the precise description of the isomorphisms). This shows that ** * is an isomorphism. We next consider two special cases of Lemma 1.7. Lemma 1.8. Fix a finite group G and a normal subgroup H C G. Fix a p-subgroup Q H, and let F : Op(G) ___! Z(p)-mod be any functor such that F (P ) = 0 whe* *never EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 9 P \ H is not G-conjugate to Q. Then the induced homomorphism GQ * G lim-*(F ) ------!~ lim- (F O Q), Op(G) = Op(N(Q)=Q) is an isomorphism, where GQ: Op(N(Q)=Q) ---! Op(G) is the functor of Lemma 1* *.7. Proof.This follows immediately from Lemma 1.7, upon letting T be the set of p- subgroups P G such that P \ H = Q. Condition (*) holds since if P \ H = Q and Q C xP x-1, then x-1Qx P \ H = Q and hence x 2 NG(Q). In the next lemma, recall that for any finite group G and any S 2 Sylp(G), a * *subgroup P S is weakly closed in S with respect to G if P is not G-conjugate to any ot* *her subgroup of S. Lemma 1.9. Fix a finite group G, a Sylow p-subgroup S 2 Sylp(G), and a subgroup Q S which is weakly closed in S with respect to G. Let F : Op(G) ___! Z(p)-m* *od be a functor such that F (P ) = 0 whenever P does not contain a subgroup G-conjuga* *te to Q. Then the induced homomorphism GQ * G lim-*(F ) ------!~ lim- (F O Q), Op(G) = Op(N(Q)=Q) is an isomorphism, where GQ: Op(N(Q)=Q) ---! Op(G) is the functor of Lemma 1* *.7. Proof.We apply Lemma 1.7, with T = {P 2 Sp(G) | Q P }. By hypothesis, P 2 T for any p-subgroup P G such that F (P ) 6= 0. For each P 2 T , there is x 2 G such that xP x-1 S; xQx-1 = Q since Q is we* *akly closed in S, and thus x 2 NG(Q). If yP y-1 Q, then xy-1 2 NG(Q) by the same reasoning, and thus y 2 NG(Q). This also shows that Q C P (the case y 2 P ), an* *d thus that condition (*) of Lemma 1.7 holds for all P 2 T . Hence lim-*(F ) ~=lim-*(F* * O GQ). The following lemma is very similar in nature to Lemma 1.7. Lemma_1.10. Fix a surjection ': G --i G0 of finite groups. Then for any functor F : Op(G0)op--! Z(p)-mod, __ __ lim-*(F O '# ) ~= lim-*(F ). Op(G) Op(G0) More generally, set H = Ker('), and let O*p(G) Op(G) be the full subcategory * *whose objects are the p-subgroups P G such that P \ H 2 Sylp(H). Then __ lim-*(F ) ~= lim-*(F ) (1) Op(G) Op(G0) __ op op for any F : Op(G0) --! Z(p)-mod and F : Op(G) --! Z(p)-mod such that __ (F O '# )|O*p(G)~=F |O*p(G), (2) and such that the action of NHP (P )=P on F (P ) is trivial for all P 2 Sp(G). * * __ Proof.It suffices to prove the last statement, since NHP (P )=P acts trivially * *on F('(P )) for each P . 10 BOB OLIVER fi For all P 2 Sp(G) such that P \ H =2Sylp(H), pfi|NHP (P )=P | (see Lemma 1.11* *), and NHP (P )=P acts trivially on F (P ). Hence by Lemma 1.4, *(NG(P )=P ; F (P )) = 0 for all such P . So by Lemma 1.2(b), lim-*(F ) ~= lim-*F |O*p(G). (3) Op(G) O*p(G) Let '0#: O*p(G) ___! Op(G0) be the restriction of '# . This is a bijection o* *n isomor- phism classes of objects, since '0#(P ) = P 0(for p-subgroups P G and P 0 G0* *) if and only if P 2 Sylp('-1P 0). It is also surjective on all morphism sets, since for* * any pair of objects P, Q in O*p(G), and any x 2 NG(P H, QH), xP x-1 QH is H-conjugate * *to a subgroup of Q 2 Sylp(QH), and hence x 2 H.NG(P, Q). Now assume x, y 2 NG(P, Q) induce the same morphism in Op(G0). After replacin* *g y by an appropriate element of Qy, we can assume that y 2 Hx = xH. Set y = xh (wh* *ere h 2 H). Then P and hP h-1 are both subgroups of x-1Qx, so [h, P ] x-1Qx \ H = P \ H (since P \ H 2 Sylp(H)), and thus h 2 N(P ). In other words, ffi Mor Op(G0)('(P ), '(Q)) ~=Mor Op(G)(P, Q) (NHP (P )=P ) . Also, NHP (P )=P has order prime to p since P 2 Sylp(HP ). The category Op(G0) is thus equivalent to O*p(G) after dividing out by the action of certain subgro* *ups of automorphisms of order prime to p. So by [BLO , Lemma 1.3], __ __ lim-*(F ) ~= lim-*(F O '0#), Op(G0) O*p(G) and (1) follows from this together with (2) and (3). 1.3___Radical_p-subgroups_ Recall that a radical p-subgroup of a finite group * *G is a p-subgroup P G such that Op(NG(P )=P ) = 1; i.e., such that NG(P )=P has no nontrivial normal p-subgroups. By Proposition 1.1(c), if P is a p-subgroup whi* *ch is not radical, then *(NG(P )=P ; M) = 0 for all Z(p)[NG(P )=P ]-modules M. Thus* * by Lemma 1.2, lim-*(F ) = 0 for any functor F on Op(G) opwhich vanishes on all rad* *ical p-subgroups of G. This helps explain the important role played by radical p-sub* *groups when working with higher limits of functors on orbit categories. In this subsec* *tion, we prove some properties of radical p-subgroups which will be needed later. We first note the following, very elementary, group theoretic lemma. Lemma 1.11. Fix subgroups P, H G such that P is a p-subgroup which normalizes H, and P \ H =2Sylp(H). Then P \ H =2Sylp(NH (P )), and hence fi pfi|NHP (P )=P | = |NH (P )=(H\ P )|. Proof.Consider the action of P on HP=P ~=H=(P \H), whose order is a multiple of p by assumption. Then (HP=P )P = NHP (P )=P ~=NH (P )=(P \H) has order a multiple of p, and thus P \ H =2Sylp(NH (P )). The next lemma lists some conditions for a p-subgroup to be radical or not. Lemma 1.12. Fix a finite group G and a prime p. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 11 Q (a)If G splitsQas a product G = i2IGi, then each radical p-subgroup P G is * *of the form P = i2IPi, where Pi Gi. (b)Let H C G be any normal subgroup. Then for each radical p-subgroup P in G, P \ H is radical in H. Conversely, if Q is radical in H, then Q = P \ H for * *some radical p-subgroup P in G. Q Proof.(a) If P G = i2IGi does not split as a product,Qthen let Pi Gi be the image of P under the projection, and set P 0= i2IPi G. Then P 0 P by assumption, and NG(P 0) NG(P ). Thus NP0(P )=P is a nontrivial normal p-subgr* *oup of NG(P )=P , and P is not p-radical. (b) Fix H C G. Assume first that P \ H is not radical in H, and set Q = Op(NH * *(P \ H)) (P \ H). Then QP P (since Q H), and so NQP (P )=P 6= 1 by Lemma 1.11. Furthermore, NQP (P ) is normalized by N(P ) since Q is, NQP (P )=P is t* *hus a nontrivial normal p-subgroup of NG(P )=P , and so P is not radical in G. Conversely, fix a radical p-subgroup Q in H, and set P = Op(NG(Q)). Then P \ H is a normal p-subgroup of NH (Q) which contains Q, and hence P \ H = Q. Clearly, NG(P ) NG(Q); they are equal since any x 2 NG(P ) normalizes P \ H = Q; and thus P is radical. 1.4___Fixed_point_and_norm_functors__P Fix a finite group G. For any subgroup H G, set NH = h2H h 2 Z[G]. For any prime p and any Z(p)[G]-module M, we consider the functors H0M, NM : Op(G) op------! Z(p)-mod, defined by setting H0M(P ) = H0(P ; M) = MP and NM(P ) = NP.M. The next proposition plays a central role _ almost as important as that of the functors *(G; M) _ when computing higher limits over orbit categories. Proposition 1.13. For any finite group G, any prime p, and any Z(p)[G]-module M, ( ( H0(G; M) = MG if i = 0 i NG.M if i = 0 lim-i(H0M) = and lim-(NM) = Op(G) 0 if i > 0 Op(G) 0 if i > 0 . Proof.Set bH0M = H0M=NM; thus bH0M(P ) = bH0(P ; M) for all P . These functors are all proto-Mackey functors in the sense of [JM ], and hence are acyclic by [* *JM , Proposition 5.14]. The complete description of the higher limits of H0M and bH0* *M is shown in [JMO , Proposition 5.2], and that of NM follows immediately. 1.5___Kan_extensions_and_limits_ If H G, and F : Op(H) op--! Ab is any func- tor, then we let F "GH: Op(G) op------! Ab denote the right Kan extension of F . To define this, fix an object P in Op(G) * *, let '#P be the overcategory whose objects are the morphisms Q ! P in Op(G) for Q H, 12 BOB OLIVER and let ~P : '#P ___! Op(H) be the forgetful functor which sends (Q ! P ) to Q* *. Then (F "GH)(P ) = lim-(F O~P), '#P and a morphism P --! P 0in Op(G) induces the obvious map between inverse limit* *s. We refer to [McL , xX.3] for more details. Lemma 1.14. The following hold for any H G and any F : Op(H) op--! Ab. (a) lim-*(F "GH) ~= lim-*(F ). Op(G) Op(H) (b)We can identify iM jPxH M (F "GH)(P ) = F (H \ gP g-1) ~= F (H \ gP g-1);(1) g2G HgP2H\G=P where (x, y) 2 P x H acts on the first sum by sending the summand for g to t* *he summand for xgy-1 via F (y). When aP a-1 Q, the induced morphism a* = (F "GH)(a): (F "GH)(Q) ------! (F "GH)(P ) satisfies (a*(,))g = F (incl)(,ga-1). Proof.By [McL , xX.3], the right Kan extension (-)"GH is a right adjoint to the* * re- striction functor from Op(G) -mod to Op(H) -mod . From this, one easily sees * *that (-)"GHpreserves exact sequences, and sends injectives in Op(H) -mod to injecti* *ves in Op(G) -mod . Also, if Z_denotes the constant functors with value Z, then lim-(F "GH) ~=Hom Op(G)-mod(Z_, F "GH) ~=Hom Op(H)-mod(Z_, F ) ~= lim-(F * *). Op(G) Op(H) Hence lim-*(F "GH) ~= lim-*(F ). Op(G) Op(H) The formulas in (b) follow immediately from the definition of (F "GH)(P ) as * *an inverse limit. In particular, the term for g 2 G corresponds to the maximal object H \ g-1 gP g-1 ___! P in the overcategory '#P . 2.Reduction to simple groups For any finite group G and any normal subgroup K C G, we define a subfunctor * *ZKG of ZG by setting ( Z(P ) \ K if P is p-centric in G ZKG(P ) = 0 otherwise. Our goal now is to study the higher limits of quotient functors ZK1G=ZK2G, when* * K1=K2 is a minimal normal subgroup of G=K2. We want to reduce the computation of high* *er limits of ZK1G=ZK2Gto the case where K1=K2 is a nonabelian simple group, K2 = Z* *(K1) is an abelian p-group, and K1 is perfect (thus a quasisimple group). This invol* *ves a series of reductions. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 13 Lemma 2.1. Fix a prime p, a finite group G, and subgroups K2 K1 both normal in G, such that K1=K2 is a minimal normal subgroup of G=K2. Set fi H = g 2 G fi[g, K1] K2 C G (thus H=K2 = CG=K2(K1=K2)), choose Q 2 Sylp(H), and set G0= NG(Q) and K0i= CKi(Q). Then K02= A x T , where A = K02\ Q = K2 \ Z(Q) is an abelian p-group and T has order prime to p. Set G00= G0=T and K00i= K0i=T. Then either (a)K01 K2 and the functor ZK1G=ZK2Gis acyclic; or (b)K001=K002~=K01=K02~=K1=K2, and K K * K0 K0 * K00 K00 lim-*ZG1=ZG2 ~= lim- ZG10=ZG20 ~= lim- ZG100=ZG200. (1) Op(G) Op(G0) Op(G00) Proof.Since K2 C H by definition, Q0def=Q \ K2 2 Sylp(K2) is p-centric in K2. T* *hus CK2(Q0) = Z(Q0) x U for some U of order prime to p. After taking fixed points o* *f the Q=Q0-action, we get that CK2(Q) = A x T , where A = CZ(Q0)(Q) = Z(Q) \ K2 and T = CU(Q). Let O*p(G) Op(G) be the full subcategory whose objects arefthoseip-subgrou* *ps P G such that P \ H 2 Sylp(H). If P \ H 2= Sylp(H), then pfi|NHP (P )=P | by Lemma 1.11, NHP (P )=P acts trivially on (ZK1G=ZK2G)(P ), and hence *(NG(P )=P ; (ZK1G=ZK2G)(P )) = 0 by Lemma 1.4. So by Lemma 1.2(b), K K * K K lim-*ZG1=ZG2 ~= lim- (ZG1=ZG2)|O*p(G). (2) Op(G) O*p(G) For any g 2 G, gQg-1 = xQx-1 for some x 2 H, since Q 2 Sylp(H), so x-1g 2 NG(Q) = G0, and g 2 HG0. Thus G = HG0. The subgroup K01K2 = CK1(Q).K2 is clearly normalized by G0= NG(Q), and is normalized by H since [H, K1] K2 by definition. Hence K01K2 C HG0= G; and the minimality assumption on K1=K2 implies that K01K2 = K2 or K1. If K01K2 = K2, then K01= CK1(Q) K2, so Z(P ) \ K1 K2 for any P Q, (ZK1G=ZK2G)|O*p(G)is the zero functor, and so lim-*ZK1G=ZK2G = 0* * by (2). This proves case (a). It remains to consider the case where K01K2 = K1, and thus where K001=K002~= K01=K02~= K1=K2. By analogy with the definition of O*p(G), let O*p(G0) Op(G0) be the full subcategory on subgroups which contain Q. Consider the functor O*p(G0) ---ff---!O*p(G) induced by the inclusion G0 G. For any P in O*p(G), P \H 2 Sylp(H), and hence * *P is conjugate to a subgroup containing Q 2 Sylp(H), and thus to an object in O*p(G0* *). If P1 and P2 both contain Q, then NG(P1, P2) = NG0(P1, P2) (any element of the transp* *orter sends Q = P1\ H = P2\ H to itself), and thus ff is the inclusion of a full subc* *ategory and an equivalence of categories. 14 BOB OLIVER For any p-subgroup P G containing Q, CG(P ) CG(Q) G0, and thus P is p-centric in G if and only if it is p-centric in G0. Also, for i = 1, 2, Z(P ) \ Ki= Z(P ) \ CKi(Q) = Z(P ) \ K0i. 0 K0 So there is a natural isomorphism (ZK1G=ZK2G)(P ) ~=(ZK1G0=ZG20)(P ) for all P * *in O*p(G0). The first isomorphism in (1) thus follows from (2), together with the analogous* * iso- morphism for G0. The second isomorphism in (1) follows from Lemma 1.10 (applied with H = T ). As a first application of Lemma 2.1, we consider the case where the subquotie* *nt K1=K2 is abelian. Proposition 2.2. Let G be a finite group. If K2 C K1 C G are subgroups, both no* *rmal in G, such that K1=K2 is abelian, then the functor ZK1G=ZK2Gis acyclic. If G is* * solvable, then the functor ZG is acyclic. Proof.Assume first that K2 C K1 C G are normal subgroups such that K1=K2 is abelian. It clearly suffices to prove the acyclicity of ZK1G=ZK2Gwhen K1=K2 is * *a minimal normal subgroup of G=K2. If K1=K2 has order prime to p, then ZK1G=ZK2G= 0, and * *so we can assume that K1=K2 is a p-group. By Lemma 2.1, we can also assume that there is a normal p-subgroup Q C G such that K2 Q, [K1, Q] = 1, and Q=K2 2 Sylp(CG=K2(K1=K2)). In particular, Q K1 since K1=K2 is an abelian p-group. So fi CG(Q) CG(K1) g 2 G fi[g, K1] K2 , and CG(Q) Q since Q is a Sylow p-subgroup of the last group. Thus Q is p-cent* *ric. Let O*p(G) Op(G) be the full subcategory with objects the p-subgroups which contain Q. If P Q, then 1 6= NPQ (P )=P Op(NG(P )=P ), so P is not radical. Thus O*p(G) contains all radical p-subgroups of G. So by Lemma 1.2 (and Proposi* *tion 1.1(c)), for any p-local functor F on Op(G) , lim-*(F ) ~=lim-*(F |O*p(G)). Set Mi = Z(Q) \ Ki. For any p-subgroup P G containing Q, Z(P ) = Z(Q)P=Q and hence ffi P=QffiP=Q (ZK1G=ZK2G)(P ) ~= Z(P ) \ K1 Z(P ) \ K2 ~=M1 M2 . Thus (ZK1G=ZK2G)|O*p(G)~=(H0M1=H0M2)|O*p(G), and hence lim-*(ZK1G=ZK2G) ~= lim-*(H0M1=H0M2). Op(G) Op(G) The last functor is acyclic by Proposition 1.13, and so ZK1G=ZK2Gis also acycli* *c. Now assume G is solvable, and let 1 = Kn C Kn-1 C . .C.K1 C K0 = G be its derived sequence; i.e., Ki+1= [Ki, Ki] for each i. We have just seen that ZKiG=* *ZKi+1G is acyclic for each i. So ZKiGis acyclic for each i, and in particular ZG = ZK* *0Gis acyclic. The following technical lemma will be useful in the later reductions. Lemma 2.3. Let G be a finite group with normal subgroups A C K C G, such that A Z(K) is an abelian p-group. Let P G be a p-subgroup such that either A * *P , EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 15 orf(Pi\ K)=A is not p-centric in K=A. Then there is a subgroup H G such that pfi|NHP (P )=P | and [H, P \ K] A; and furthermore K A * NG(P )=P ; (ZG =ZG )(P ) = 0. fi Proof.Set H = x 2 K fi[x, P \ K] A . Then P normalizes H. Also, P \ H =2 Sylp(H): this is clear if A P ,fandifollows by definition (of p-centric) if (* *P \ K)=A is not p-centric in K=A. Hence pfi|NHP (P )=P | by Lemma 1.11. Since NHP (P )=P* * acts trivially on Z(P)\K_Z(P)\A, the last statement now follows from Proposition 1.1* *(b). We next reduce the computation of lim-*(ZK1G=ZK2G) when K1=K2 is nonabelian to the case where K1 is quasisimple; i.e., to the case where K1=K2 is nonabelian a* *nd simple, K2 = Z(K1), and K1 is perfect. Lemma 2.4. Fix a finite group G, and subgroups A K G, both normal in G, such that A is a p-group, [A, K] = 1, and K=A is nonabelianQand a minimal normal sub* *group of G=A. Then A = Z(K), and we can write K=A = j2JLj, where each Lj is simple and a minimal normal subgroup of K=A. Let Kj K be such that Lj = Kj=A, and set eLj= [Kj, Kj], Aj = eLj\ A = Z(eLj), and Gj = NG(eLj). Then for any given j 2 J, Lej Aj lim-i(ZKG=ZAG) ~= lim-i(ZGj=ZGj) Op(G) Op(Gj) for all i 2, and there is a surjection Lej Aj 1 K A lim-1(ZGj=ZGj) --- --i lim-(ZG =ZG ). Op(Gj) Op(G) Proof.Since K=A is nonabelian and a minimal normal subgroup of G=A, it must be a product of nonabelian simple groups isomorphic to each other (cf. [Gor , The* *orem 2.1.5]).QSince we also assume [A, K] = 1, this implies that A = Z(K). So write * *K=A = j2JLj where Lj = Kj=A as above. For any i 6= j, Ki\ Kj = A, and hence [Ki, Kj] A. Since A is central, this means that the commutator map [-, -]: Kix Kj --! A is a homomorphism in each coordinate; e.g., [gg0, h] = [g, h][g0, h] for g, g0 * *2 Ki and h 2 Kj. Thus [Ki, Kj] = 1 for all i 6= j, (1) since Ki=A and Kj=A are simple and centralize A. Lej Aj op Set Zj = ZGj=ZGj for short. Define functors 1, 2: Op(G) ___! Ab by setti* *ng iY jP 1(P ) = Zj(P \ Gj) and 2 = ZKG=ZAG. j2J Each factor Zj(P \ Gj) in 1(P ) can be identified with a subgroup of Lj ~=eLj=* *Aj, and hence 1(P ) can be identified with a subgroup of K=A. Under this identificatio* *n, 1 sends the morphism in Op(G) represented by x 2 N (P, Q) to that given by restri* *ction of x-1. Let 01and 02be the functors ( i(P ) if P \ K is p-centric in K 0i(P ) = 0 otherwise. 16 BOB OLIVER In Step 1, we show that lim-*( 1) ~= lim-*(Zj) for all j 2;J (2) Op(G) Op(Gj) and in Step 2 that lim-*( 01) ~= lim-*( 1) and lim-*( 02) ~= lim-*( 2) (3) Op(G) Op(G) Op(G) Op(G) In Step 3, we identify 01as a subfunctor of 02, and prove in Step 4 that 02=* * 01is acyclic. The lemma then follows from the relative exact sequence for higher lim* *its of the pair 01 02. Step 1: For each j 2 J, there is an obvious morphism of functors on Op(Gj) 1|Op(Gj)-----! Zj, defined by projection to the j-th factor, which is adjoint to a natural morphism ! : 1 -----! Zj"GGj. Since G acts transitively on the factors Lj K=A (otherwise K=A would not be a minimal normal subgroup), ! is an isomorphism of functors by the formula in Lem* *ma 1.14(b) for Kan extensions. Hence (2) follows from Lemma 1.14(a). Step 2: To prove (3), it suffices using Lemma 1.2(b) to show that *(NG(P )=P ; i(P )) = 0 whenever i = 1, 2 and P \ K is not p-centricfiniK. By Proposition 2.3, there is H G such that [H, P \ K] A and pfi|NHP (P )=P |. Also, both groups i(P ) c* *an Q be identified with subgroups of K=A ~= j2JeLj=Aj, so NHP (P )=P acts triviall* *y on i(P ), and *(NG(P )=P ; i(P )) = 0 by Lemma 1.4. Q Q Step 3: Set Ke = j2JeLjand Ae = j2JAj. We will write Rp for the Sylow p- subgroup of a group R, but only in situations where it is a direct factor of R * *and abelian. Fix a p-subgroup P G such that P \ K is p-centric in K, and note that hY iP hY iP CKe(P ) ~= CeLj(P \ Gj) and CAe(P ) ~= CAj(P \ Gj) . j2J j2J Furthermore, the action of any g 2 P permutes the factors under each of these p* *roduct decompositions, and is trivial whenever it sends a factor to itself. Hence " #P Q P Y CeLj(P \ Gj)p j2JCeLj(P \ Gj)p ffi 1(P ) ~= ____________ ~=____________________QP~=CKe(P )p CAe(P ). j2J CAj(P \ Gj) j2JCAj(P \ Gj) We can thus write ffi 0 ffi 01(P ) ~=CKe(P )p CAe(P ) and 2(P ) ~=CK (P )p CA(P ) for all P such that P \ K is p-centric in K. The natural map from eK to K induc* *es a natural morphism of functors 01___! 02, and this is injective since 01(P ) * *and 02(P ) can both be identified as subgroups of K=A ~=eK=Ae. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 17 Step 4: Set F = 01= 02for short; it remains to show that F is acyclic. Fix * *S 2 Sylp(K). Define functors __ op F0: Op(G) op-----! Z(p)-mod and F: Op(NG(S)=S) -----! Z(p)-mod by setting ( F (P )if P \ K 2 Sylp(K) __ F0(P ) = and F(P=S) = F (P ) = F0(P ). 0 otherwise We claim that __ lim-*(F ) ~= lim-*(F0) ~= lim-* (F ), (4) Op(G) Op(G) Op(NG(S)=S) __ and that F is acyclic. The second isomorphism follows immediately from Lemma 1.* *8. Q Recall that K=A = j2JKj=A, where the factors Kj=A are simple. Also, eLj= [Kj, Kj] is perfect (hence quasisimple), with center Z(eLj) = Aj = eLj\ A. To p* *rove the first isomorphism in (4), it will suffice, by Lemma 1.2, to show that *(NG(P )=P ; F (P )) = 0 whenever P \ K =2Sylp(K). (5) Q Let Q be the set of p-subgroups Q K such that Q A, and Q=AQ= j2JQj=A for some Qj Kj. Since A G is a normal p-subgroup and K=A = j2JKj=A, P \K 2 Q for all radical p-subgroups P G by Lemma 1.12(a,b). Hence if P \ K =2Q, then P is not radical, and (5) holds by Proposition 1.1(c). Now let P G be any p-subgroup such that Q = P \ K 2 Q and is p-centric in K. The following sequence is exact 1 ___! CAe(P ) ___! CKe(P )p x CA(P ) ___! CK (P )p ___! F (P )(___!* *6)1 since 02(P ) ~=CKe(P )p=CAe(P ) and (ZKG=ZAG)(P ) ~=CK (P )p=CA(P ) by Step 3. We claim that the sequence 1 ___! eA___! CKe(Q) x A ___! CK (Q) ___! 1 (7) is exact. This is obtained from the exact sequence 1 ! Ae! Ke x A ! K ! 1 by taking fixed subgroups of the conjugation action of Q. So the exactness of (7)* * will follow upon showing that CKe(Q) x A surjects onto CK (Q). Any g 2 K can be writ* *ten Q g = a. j2Jgj for a 2 A and gj 2 eLj, [g, Q] = 1 implies [gj, Qj] = 1 for each * *j (since [Qj, eLi] [Kj, Ki] = 1 for i 6= j by (1)), and thus ga-1 is the image of (gj)* *j2J 2 CKe(Q). After taking fixed points of the P=Q-action on (7), we get an exact sequence 1 ___! CAe(P ) ___! CKe(P )p x CA(P ) ___! CK (P )p ___! H1(P=Q; eA). A comparison with (6) shows that F (P ) is contained in H1(P=Q; eA) as a module* * over N(P )=P . In particular, if Q = P \ K =2Sylp(K), then NKP (P )=P has order a mu* *ltiple of p (Lemma 1.11), and acts trivially on F (P ) since it acts trivially on P=Q * *~=P K=K and on eA. So (5) follows from Lemma 1.4 in this case. Finally, when Q = P \ K = S, the first three terms in the exact sequence (6) * *are acyclic as functors on Op(NG(S)=S) by Proposition 1.13.__(For example, CKe(P )* * = CKe(S)P=S is the fixed subgroup of the P=S-action.) So F is also acyclic, and* * this finishes the proof that 01= 02is acylic. 18 BOB OLIVER We have now reduced the computation of lim-*(ZK1G=ZK2G) to the case where K1 * *is quasisimple and K2 = Z(K1) is an abelian p-group. It turns out that this only d* *epends on K1 and AutG (K1). To make this precise, we define certain functors YL as fol* *lows. As usual, cx denotes conjugation by an element x. Definition 2.5. Fix a finite group L and a group of automorphisms Aut(L) wh* *ich contains Inn(L). Define YL : Op( )op ___! Ab by setting ( fi cx 2 Inn(L) fix 2 CL(P ) pif P \ Inn(L) is p-centric in Inn(L) YL(P ) = 0 otherwise. For any P, Q , YL sends the morphism represented by g 2 N (P, Q) to conjugat* *ion by g-1. Here, we again write Kp for the Sylow p-subgroup of K in a situation where it* * is a direct factor of K and abelian. Thus whenever P \ Inn(L) is p-centric in Inn* *(L), YL(P ) CInn(L)(P \ Inn(L))p = Z(P \ Inn(L)). Lemma 2.6. Fix a finite group G with quasisimple normal subgroup L C G, and assume that A = Z(L) is a p-group. Set = AutG (L). Then lim-*(ZLG=ZAG) ~= lim-*(YL). Op(G) Op( ) Proof.Let Z0: Op(G) op___! Ab be the functor Z0(P ) = (ZLG=ZAG)(P ) if P A a* *nd (P \ L)=A is p-centric in L, and Z0(P ) = 0 otherwise. We regard this as a quot* *ient functor of ZLG=ZAG. By Proposition 2.3, *(N(P )=P ; (ZLG=ZAG)(P )) = 0 for any* * P such that Z0(P ) = 0, and hence lim-*(ZLG=ZAG) ~=lim-*(Z0) by Lemma 1.2(a). It remains to show that lim-*(Z0) ~=lim-*(YL). Let c: G -i be the surjecti* *on which sends g 2 G to cg 2 Aut(L). Set H = Ker(c) = CG(L), and let O*p(G) Op(G) be t* *he full subcategory whose objects are the p-groups P G such that P \ H 2 Sylp(H). By Lemma 1.10, it suffices to show that (a)(YL Oc# )|O*p(G)~=Z0|O*p(G); and (b)the action of NHP (P )=P on ZLG=ZAGis trivial for all p-subgroups P G. Point (b) is immediate from the definition of the functor ZKG. Also, for any P* * G such that P \ H 2 Sylp(H), ( CL(P )p=CL(A) if (P \ L)=A is p-centric in L=A (YL Oc# )(P ) = YL(Aut P(L))~= 0 otherwise ~= Z0(P ), and this proves (a). Lemma 2.6 finishes the process of reducing the general computation of lim-*(Z* *G) to that of lim-*(YL) when L is quasisimple and Inn(L) Aut(L). These results * *are now summarized in the following: Proposition 2.7. Fix a finite group G and normal subgroups K2 K1 in G, such t* *hat K1=K2 is a minimal subgroup of G=K2. If K1=K2 is abelian, or if there is a p-su* *bgroup Q G such that [Q, K1] K2 and CK1(Q) K2, then the functor ZK1G=ZK2Gis EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 19 acyclic. Otherwise, there is a quasiperfect group L with p-group center such th* *at K1=K2 is isomorphic to a product of copies of L=Z(L), a subgroup Aut (L) containi* *ng Inn(L), and (for each i) a homomorphism K K lim-i(YL) -----! lim-iZG1=ZG2 Op( ) Op(G) which is onto when i = 1 and an isomorphism when i 2. Proof.The abelian case was handled in Proposition 2.3. The cases where K1=K2 is nonabelian follow from Lemmas 2.1, 2.4, and 2.6. In order to avoid repeating these conditions about central extensions and gro* *ups of automorphisms throughout the remaining sections, we define the following classe* *s of finite simple groups. Definition 2.8. For each prime p and each i 1, let Li(p) be the class of fini* *te nonabelian simple groups L with the property that lim-i(YeL) = 0 Op( ) for each quasisimple group eLsuch that Z(eL) is a p-group and eL=Z(eL) ~=L, and* * each subgroup Aut(eL) which contains Inn(eL). Also, L i(p) denotes the intersect* *ion of the classes Lj(p) for all j i. The important consequence of Proposition 2.7 is: Proposition 2.9. For any finite group G and any i 1, lim-i(ZG) = 0 if for each nonabelian simple group L which appears in the decomposition series for G, L 2 * *Li(p). Our goal now, throughout the rest of the paper, is to show that every finite * *non- abelian simple group lies in L 2(2). 3.A relative version of -functors Throughout this section, p denotes a fixed prime. We first present a relative* * version of the functors *(G; M) which will be useful when handling outer automorphisms* * of a quasisimple group L. Definition 3.1. Fix a prime p, a pair H C G of finite groups, and a Z(p)[G]-mod* *ule M. Let G,HM: Op(G) op--! Z(p)-mod be the functor defined by ( MP if P \ H = 1 G,HM(P ) = 0 otherwise; and define *(G, H; M) = lim-*( G,HM). Op(G) Note in particular that GM= G,GM, and hence *(G; M) = *(G, G; M). 20 BOB OLIVER The importance of these groups arises from the following generalization of [J* *MO , Proposition 5.4]. For any H C G, we say that a functor F : Op(G) op___! Ab is * *H- controlled if for all P , the inclusion of (P \H) in P induces an isomorphism F* * (P ) ~= F (P \H)P. Proposition 3.2. Fix a finite group G, a prime p, a normal subgroup H C G, and a p-subgroup Q H. Let F : Op(G) --! Z(p)-mod be any H-controlled functor which vanishes except on subgroups P G such that P \ H is G-conjugate to Q. (Thus F (P ) = F (Q)P=Q whenever P \ H = Q.) Then lim-*(F ) ~= * NG(Q)=Q, NH (Q)=Q; F (Q) . Op(G) Proof.This is a special case of Lemma 1.8. The idea now is to filter an arbitrary H-controlled functor F : Op(G) ___! Z* *(p)-mod by subfunctors Fiin such a way that for each quotient functor Fi=Fi-1, Fi=Fi-1(* *P ) = 0 except for those P G such that P \ H lies in one conjugacy class of p-subgrou* *ps of H. Then each lim-*(Fi=Fi-1) is described via Proposition 3.2 and the *(G, H* *; -). When L is quasisimple and Inn(L) Aut(L), the functors YL need not be Inn(* *L)- controlled, but we will see that the same techniques can be used to compute hig* *her limits of these functors. For any G, and any short exact sequence of Z(p)[G]-modules 0 --! M0 -----! M -----! M00--! 0, the induced sequence of functors 0 --! GM0--! GM--! GM00--! 0 is also exact, and hence induces a long exact sequence of the groups *(G; -). This is not in * *general the case for the relative groups *(G, H; -) when H C G is a proper normal subg* *roup, which is why the statement of point (b) in the next proposition is so detailed. Most of the properties of the relative groups *(G, H; M) listed in the follo* *wing proposition generalize properties of the groups *(G; M) proven in [JMO ]. Proposition 3.3. Fix a prime p, a finite group G, a normal subgroup H C G, and a Z(p)[G]-module M. Then the following hold. (a)Iff(p,i|H|) = 1, then 0(G, H; M) ~=MG , and i(G, H; M) = 0 for all i > 0. * * If pfi|H|, then 0(G, H; M) = 0. (b) *(G, H; M) = 0 if the kernel of the action of H on M has order a multiple o* *f p. More generally, if M0 M is a G-invariant submodule, then o *(G, H; M) ~= *(G, H; M0) if the kernel of the H-action on M=M0 has ord* *er a multiple of p; and o *(G, H; M) ~= *(G, H; M=M0) if the kernel of the H-action on M0 has ord* *er a multiple of p. (c) *(G, H; M) = 0 if Op(H) 6= 1. (d)If K C G is a normal subgroup which acts trivially on the Z(p)[G]-module M, * *and H \ K has order prime to p, then *(G, H; M) ~= *(G=K, HK=K; M). EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 21 fi (e)Assume pfi|H|, let ~ be the equivalence relation on Sylp(H) generated by non* *trivial intersection, and set G0 = {g 2 G | gSg-1 ~ S} for some fixed S 2 Sylp(H). T* *hen 1(G, H; M)~=MG0=MG . (f)Assume the Sylow p-subgroups of H are cyclic, or (if p = 2) quaternion. Then i(G, H; M) = 0 for all i 2. Proof.(a) If (p, |H|) = 1, then P \ H = 1 for all p-subgroups H G, and hence G,HM= H0M in the notation of Propositionf1.13.iSo by that proposition, lim-0( * *G,HM) = MG and its higher limits vanish. If pfi|H|, then G,HMvanishes on the Sylow sub* *groups of G, and hence lim-0( G,HM) = 0. (b) The first statement is a special case of either of the other two. Assume first that K def=Ker[H ___! Aut(M=M0)] C G has order a multiple of p. Fix P G such that P \ H = 1, and choose Q 2 Sylp(P K) such that Q P . Then NQ(P ) P , so NQ(P )=P 6= 1 acts trivially on MP =MP0, and hence G,H G,H * NG(P )=P ; ( M = M0 )(P ) = 0 by Proposition 1.1(b). Since this holds for all such P , lim-*( G,HM= G,HM0) = * *0 by Lemma 1.2(a), and hence *(G, H; M) ~= *(G, H; M0). Now assume that K def=Ker[H ___! Aut(M0)] C G has order a multiple of p. The* *re is an exact sequence of functors on Op(G) 0 ---! G,HM0-----! G,HM-----! G,HM=M0-----! ---! 0, where (P ) H1(P ; M0) if P \ H = 1 and (P ) = 0 otherwise. We have just seen that lim-*( G,HM0) = 0, and hence we will be done upon showing that lim-*( ) = * *0. For each P G such that P \ H = 1, NPK (P )=P has order a multiple of p (Lemma 1.11) and acts trivially on (P ) H1(P ; M0), and thus *(NG(P )=P ; (P )) =* * 0 by Proposition 1.1(b). So all higher limits of vanish by Lemma 1.2(a). (c) If Op(H) 6= 1, then for all p-subgroups P G such that P \ H = 1, Op(NG(P )=P ) NOp(H).P(P )=P 6= 1. Hence *(NG(P )=P ; MP ) = 0 for all such P by [JMO , Proposition 6.1(ii)], an* *d so *(G, H; M) = 0 by Lemma 1.2. (d) Let ': G --! G=K be the projection, and let '# denote the induced functor between orbit categories. Let O*p(G) Op(G) be the full subcategory whose obje* *cts are those P G such that P \ K 2 Sylp(K). We claim that G=K,HK=K G,HM|O*p(G)~= M O'# |O*p(G). (1) This is clear, once we have checked that for any p-subgroup P G such that P \* * K 2 Sylp(K), P \ H = 1 if and only if P K=K \ HK=K = 1. If P K=K \ HK=K = 1, then P \ H K, and hence P \ H = 1 since P is a p-group and K \ H has order prime to p. Conversely, if P \ H = 1, then P K \ H has order prime to p, since P 2 Sylp(P* * K) and thus any element of P K \ H of p-power order would be G-conjugate to an ele* *ment of P \ H. Hence P K \ H K, since any element of P Kr K has order a multiple of p. It follows that P K \ HK K, and P K=K \ HK=K = 1. 22 BOB OLIVER This finishes the proof of (1). Lemma 1.10 now applies to show that *(G, H; M) = lim-*( G,HM) ~= lim-*( G=K,HK=KM) = *(G=K, HK=K; M). Op(G) Op(G=K) (e) Regard G,HMas a subfunctor of the functor H0M, which sends P to MP for all P . By Proposition 1.13, H0M is acyclic and lim-0(H0M) ~=MG . So there is a sho* *rt exact sequence 0 --! MG -----! lim-0(H0M= G,HM) -----! 1(G, H; M) --! 0. It remains only to show that the middle term is isomorphic to MG0. To see this, fix some T 2 Sylp(G) which contains our given S 2 Sylp(H), and r* *estrict to the full subcategory OT(G) Op(G) whose objects are the subgroups of T . Th* *en G0 = {g 2 G | gSg-1 ~ S}= = ; and hence lim-0(H0M= G,HM) ~=MG0. (f) Fix bS2 Sylp(G), set S = bS\ H 2 Sylp(H), and let Z S be the unique subg* *roup of order p. In particular, Z is weakly closed in bSwith respect to G. Regard G,HMas a subfunctor of the functor H0M which sends P to MP for all P . Then H0M= G,HMsends a p-subgroup P G to MP if P contains a subgroup conjugate to Z and sends P to 0 otherwise. So by Lemma 1.9, P lim-*(H0M= G,HM) ~= lim-* P=Z 7! M , Op(G) Op(NG(Z)=Z) and this last functor is acyclic by Proposition 1.13. Since H0M is also acycli* *c by Proposition 1.13, the exact sequence for the extension of functors now shows th* *at i(G, H; M) = lim-i( G,HM) = 0 for all i 2. The next proposition describes the role of these relative functors *(G, H; M* *) when computing higher limits of the YeL. For these purposes, it is useful to consid* *er the following subquotient functors of YeL, defined for any p-centric subgroup Q L: ( Ye(P ) if P Q0, some Q0 -conjugate to Q (YeL) Q (P=) L 0 otherwise ( Ye(P ) if P \ L is -conjugate to Q (YeL)Q(P )= L 0 otherwise Thus (YeL)Q is a subfunctor of (YeL) Q , and this is a quotient functor of YeL.* * When and eLare clear from context, we drop them, and just write YQ Y Q , etc. i Lemma 3.4. Fix a simple group L, a central extension eL- -i L such that eL is quasisimple and Ker (ß) is a p-group, and a subgroup Aut (eL) which contains Inn(eL) ~=L. Then the following hold for any p-centric subgroup Q L. (a)If Q =2Sylp(L), then lim-*((YeL)Q) ~= *(N (Q)=Q, NL(Q)=Q; YeL(Q)). Op( ) EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 23 (b)If S 2 Sylp(L), and Q is p-centric in L and weakly closed in S with respect * *to , then for all i 0, ( (ß-1Q)N (Q)=Z(eL) if i = 0 lim-i((YeL) Q ) ~= Op( ) 0 if i > 0. Proof.Let F 1 F 0be the following functors on Op( ): ( (ß-1(P \ L))P if P \ L is p-centric in L 0 P F 1(P ) = and F (P ) = Z(eL) . Z(eL)P otherwise Here, we identify L = Inn(eL) C . By definition, Y = YeL~=F 1=F 0. Define quot* *ient functors F iQof F iand subfunctors FQi F iQin analogy with the definitions for* * Y . Thus Y Q ~=F 1Q=F 0Q and YQ ~=FQ1=FQ0 By Proposition 3.2, i lim-*(FQi) ~= * N (Q)=Q, NL(Q)=Q; FQ(Q) Op( ) for i = 0, 1. Since the action of NL(Q)=Q on F 0(Q) = Z(eL) is trivial, Propos* *ition 3.3(b) implies that if Q =2Sylp(L), then 1 0 lim-*(YQ) ~= lim-*(FQ1=FQ0) ~= * N (Q)=Q, NL(Q)=Q; F (Q)=F (Q) Op( ) Op( ) ~= * N (Q)=Q, NL(Q)=Q; Ye(Q) . L Now assume that Q is p-centric in L, and weakly closed in some S 2 Sylp(L) wi* *th respect to . By definition, for all P Q in , F i(P ) = F i(Q)P for i = 0, 1* *. Hence by Lemma 1.9 and Proposition 1.13, ( 0 if * > 0 lim-*(F iQ) ~= F i(Q)N (Q) if * = 0. Thus Y Q ~=F 1Q=F 0Qis acyclic, and lim-0(Y Q ) ~=(ß-1Q)N (Q)=Z(eL) . We will need a much stronger vanishing theorem for the *(G, H; M) than what was shown in Proposition 3.3. In the following lemma, for any set H1, . .,.Hm * * G of subgroups, we write NG(H1, . .,.Hm ) to denote the subgroup of elements of G wh* *ich normalize all of them. A radical p-chain of length n in G is a sequence Op(G) = P0 C P1 C . .C.Pn of distinct p-subgroups of G such that Pi= Op(NG(P0, . .,.Pi)) for all i (in pa* *rticular, PiC Pn for all i), and such that Pn 2 Sylp(NG(P0, . .,.Pn-1)). Proposition 3.5. Fix a finite group G, a normal subgroup H C G, and a finite Z(* *p)[G]- module M. Assume, for some n 1, that n(G, H; M) 6= 0. Let 0 = M00 M01 . . .M0n= M=pM be any filtration by Z(p)[G]-submodules. Then there is a radical p-chain 1 = P0 P1 P2 . . .Pn 24 BOB OLIVER of length n in H such that NNH(P1,...,Pn).(M0k=M0k-1) 6= 0 for some k, and such* * that rkp(M) rk(M0k=M0k-1) |Pn| pn. Proof.For any Z(p)[G]-module M, let MGM be the set of all sub-Mackey functors H0M; i.e., those functors on Op(G) such that (P ) MP for all P , and such * *that the relative norm NPQ: MQ --! MP sends (Q) into (P ) for all p-subgroups Q * * P in G. Let NMG,Hbe the set of all functors on Op(G) of the form \ G,HM(object* *wise intersection) for some 2 MGM. We first show: for any 2 NMG,Hand any n 1 with lim-n( ) 6= 0, there exists a radical * *p- subgroup P H such that NP.M 6= 0, and a functor 02 NNNG(P)=P,NH(P)=PP.M* *(1) such that lim-n-1( 0) 6= 0. To see this, let 0 2 MGM be such that = 0\ G,HM. Let 0 be the subfunc* *tor (P ) = 0(P ) \ (NP\H .M); this still lies in MGM. Since is acyclic [JM , Pr* *oposition 5.14], lim-n-1( = ) 6= 0. By an appropriate filtration of = , there is a p-su* *bgroup 1 6= P H such that if we define P by P(Q) = (Q) if Q \ H is conjugate to P and 0(Q) = 0 otherwise (still as a functor on Op(G) ), then lim-n-1( P) 6= 0* *. By Proposition 1.1(a,c), for some p-subgroup Q G such that Q \ H = P , Q is radi* *cal in G, and hence P is radical in H by Lemma 1.12(b). Furthermore, if we define 0 0on Op(NG(P )=P )by setting 0(Q=P ) = (Q) and 0(Q=P ) = P(Q), then 02 MNG(P)=PNP.M, and so 0= 0\ NG(P)=P,NH(P)=PNP.M2 NNNG(P)=P,NH(P)=PP.M. Finally, by Lemma 1.8, the functors P and 0 have the same higher limits, and * *in particular lim-n-1( 0) 6= 0. This finishes the proof of (1). We next claim, by induction on n, that for any 2 NMG,Hwith lim-n( ) 6= 0, there exists a radical p-chain 1(=2P0) P1 . . .Pn of length n such that NK .M 6= 0, where K = NH (P1, . .,.Pn). To prove this, let P and 0be as in (1). If n = 1, then 0 6= lim-0( 0) 0(NG(P )=P, NH (P )=P ; NP.M) implies that (p, |NH (P )=P |) = 1, hence that P 2 Sylp(H), and (NP.M)NH(P) = NNH(P).M 6= 0 by Proposition 3.3(a). Thus (0 P ) is a radical p-chain, and (2) holds in thi* *s case. If n > 1, then by the induction hypothesis (applied to 0), there is a radica* *l p-chain 1 6= P2=P . . .Pn=P in NH (P )=P (3) such that NK=P.(NP.M) = NK .M 6= 0, where K=P = NNH(P)=P(P2=P, . .,.Pn=P ). Since P is radical in H, Op(NH (P )=P ) = 1, and the sequence 1 P P2 . . * *.Pn is a radical p-chain of length n in H. Also, K = NH (P, P2, . .,.Pn), and this * *finishes the proof of (2). Now let M be any finite Z(p)[G]-module such that n(G, H; M) = lim-n( G,HM) 6* *= 0, and let 0 = M00 M01 . . .M0n= M=pM be a filtration by G-invariant submodules. Since each pkM=pk+1M is isomorphic to an Fp[G]-submodule of M=pM, we can lift this to a filtration of M by submodules M00isuch that each M00i=M00i-1is isomor* *phic EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 25 G,H G,H to a submodule of some M0j=M0j-1. Choose ` such that lim-n M00`= M00`-16= 0. * *Set V = M00`=M00`-1for short. Consider the functor : Op(G) op--! Z(p)-mod defined by (P ) = (M00`)P=(M00`* *-1)P. This is a sub-Mackey functor of H0V , and thus an element of MGV. Hence def= \ G,HV= G,HM00`= G,HM00`-12 NVG,H. By (2), since lim-n( ) 6= 0, there is a radical p-chain 1 = P0 P1 . . .Pn o* *f length n such that NK .V 6= 0, where K = NH (P1, . .,.Pn). In particular, NPn.V 6= 0. Fix x 2 V such that NPn.x 6= 0, and consider the P* *n-linear homomorphism Fp[Pn] -ff-!V defined by setting ff(g) = gx. Thus Ker(ff) is an id* *eal in Fp[Pn] which does not contain NPn, and hence is the zero ideal (cf. [Se, x8.* *3, Prop. 26]). So ff is injective, and V contains a copy of Fp[K]. Since V is isomor* *phic to a subgroup of some M0k=M0k-1, this module also contains a copy of Fp[Pn]. The lo* *wer bounds for rkp(M) are now immediate. 4.Subgroups which contribute to higher limits As noted earlier, our remaining goal is to prove that every nonabelian finite* * simple group lies in the class L 2(2) (see Definition 2.8). Once we have shown this, * *then Theorem A will follow as a consequence of Proposition 2.9. The aim of this sect* *ion is to prove a series of propositions, whose hypotheses are stated in purely group * *theoretic terms (without reference to higher limits or *'s), which can then be used in l* *ater sections to carry out a case-by-case check that L 2 L 2(2). The following definition is mostly of interest when L is simple, but in a few* * cases (when carrying out inductive arguments) we need to also deal with almost simple groups. For simplicity in notation, for any group L with Z(L) = 1, we identify * *L = Inn(L) as a subgroup of Aut(L). Recall that for any p-group P and any n 1, n n(P ) def=. fi Definition 4.1. Fix a centerfree group L and a prime pfi|L|. (a)For each i 1, let Ri(L ; p) be the set of all p-subgroups P L with the p* *rop- erty that for some Aut(L) which contains Inn(L), and some N (P )-invaria* *nt subgroup Z0 Z(P ), i(N (P )=P, NL(P )=P ; Z0) 6= 0. (b)For each i 1, set fi i Ei(L ; p) = 1(Z(P )) fiP 2 R (L ; p) . (c)An elementary abelian p-subgroup E L is called pivotal if E = 1(Z(P )) for some P 2 Sylp(CL(E)), and Op(Aut L(E)) = 1. S We also write R i(L ; p) = j iRj(L ; p), and similarly for E i(L ; p). In a* *ddition, for any p-subgroup Q L, we let Ri(L ; p) Q and R i(L ; p) Q denote the set * *of subgroups in Ri(L ; p) and R i(L ; p), respectively, which do not contain any s* *ubgroup conjugate to Q. 26 BOB OLIVER As will be seen in the next proposition (or in its proof), we can think of Ri* *(L ; p) as the set of p-subgroups of L which could öc ntribute" to lim-i(YeL), for some qu* *asicentric group eLwith eL=Z(eL) ~= L, and some Aut (eL) which contains Inn(eL) ~= L. * * Of course, the existence of an element of Ri(L ; p) does not mean that lim-i(YeL) * *is non- vanishing for some and eL: a nonzero öc ntribution" to lim-i(-) by a subgrou* *p in Ri(L ; p) could be cancelled by a contribution to lim-i(1-). Proposition 4.2. Fix a prime p, a finite simple group L, S 2 Sylp(L), and i 1. Assume there is a subgroup Q S which is p-centric in L and weakly closed in S with respect to Aut (L), such that Ri(L ; p) Q = ?. Then L 2 Li(p). In partic* *ular, L 2 Li(p) if Ri(L ; p) = ?. Proof.Fix and eL, and let Y Q be the subfunctor of YeLdefined by setting Y Q * *(P ) = YeL(P ) if P does not contain a subgroup conjugate to Q and Y Q (P ) = 0 otherw* *ise. For each P L ~= Inn(eL) which does not contain a subgroup conjugate to Q, we defined YeL(P ) to be a certain N (P )=P -invariant subgroup of Z(P ) (Definiti* *on 2.5). Thus P =2Ri(L ; p) implies that i(N (P )=P, NL(P )=P ; YeL(P )) = 0. Hence lim-i(Y Q ) = 0 by Lemma 3.4(a) and an appropriate filtration of Y Q . F* *ur- thermore, lim-i(YeL=Y Q ) = 0 by Lemma 3.4(b), and this finishes the proof of t* *he proposition. The last statement is just the case Q 2 Sylp(L). In the course of the next five sections, we will prove that all simple groups* * lie in L 2(2), and most cases this will be done using Proposition 4.2. The following p* *roposi- tion will, however, be needed when proving that simple groups of Lie type in ch* *arac- teristic two lie in L 2(2) (and in fact, with the exception of L3(2), lie in L * *1(2)). Proposition 4.3. Let L be a simple group. Fix S 2 Sylp(L), and let Q C S be a p-subgroup with the following properties: (a)Q is p-centric in L and weakly closed in S with respect to Aut(L). (b)for each P S which does not contain Q, and which is p-centric and radical * *in L, P is weakly closed in S with respect to L, and Z(NL(P ))p = Z(NL(P Q))p. Then L 2 L 1(p). Proof.Let P be the set of p-subgroups of S which are radical and p-centric in L* *. For any P 2 P, let ePdenote its inverse image in eL. We first check that for each P* * 2 P, Z(NeL(Pe))p = Z(NeL(gP Q))p. (1) By assumption, Z(NL(P ))p = Z(NL(P Q))p. Also, _P Z(NeL(Pe))=Z(eL) = Ker Z(NL(P )) ---! Hom (NL(P ), Z(eL)) , and similarly Z(NeL(gP Q))=Z(eL) = Ker(_PQ ). Here, _P and _PQ are defined by l* *ifting to eLand taking commutators. Thus _P and _PQ have the same domain. Since Z(eL) * *is a p-group and NL(P ) NL(P Q) S (P and Q being weakly closed in S), their ta* *rget EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 27 groups both inject into Hom (S, Z(eL)). Thus Ker(_P) = Ker(_PQ ), and this pro* *ves (1). For each R 2 P, define functors YR and Y R on Op( ) by setting ( Ye(P ) if P \ L R0, some R0 -conjugate to R Y R (P ) = L 0 otherwise and ( Ye(P ) if P \ L is -conjugate to R YR(P ) = L 0 otherwise. Thus Y R is a quotient functor of YeL, and YR is a subfunctor of Y R . We claim that lim-*(YR) = 0 for each R 2 P which does not contain Q. It suffi* *ces to prove this when N (R).L = (otherwise replace by N (R).L without changing the higher limits). In this case, R and RQ are both weakly closed in S with respect* * to . Hence by Lemma 3.4(b), Y R and Y RQ are acyclic, and by (1), ffi =Lffi 0 lim-0(Y R ) ~=Z(NeL(Re))p=L Z(eL) ~=Z(NeL(gRQ))p Z(eL) ~=lim-(Y RQ ).(2) Thus if we set Yo R = Ker[Y R ___! Y RQ ], then lim-*(Yo R) = 0. We can assume inductively that lim-*(YP) = 0 for all P 2 P such that P R and P Q. Via the obvious filtration of Yo R, we now see that lim-*(YR) = 0. Thus lim-*(YeL) ~=lim-*(Y Q ), and Y Q is acyclic by Lemma 3.4(b) again. Sinc* *e this holds for all and eL, L 2 L 1(2). It remains to develop some tools for determining which subgroups belong to th* *e sets Ri(L ; p). The first step is to study their connection with the sets Ei(L ; p)* * and the pivotal subgroups of L, also defined in 4.1. Proposition 4.4. Fix a prime p, a finite centerfree group L, and i 1. Then t* *he following hold for any p-subgroup P L and any elementary abelian p-subgroup E* * L. (a)If P 2 Ri(L ; p) and E = 1(Z(P )), then P 2 Sylp(CL(E)). If E 2 Ei(L ; p) a* *nd P 2 Sylp(CL(E)), then P 2 Ri(L ; p) and E = 1(Z(P )). In other words, there are inverse bijections elem Ri(L ; p)=(conj) --------!--------Ei(L ; p)=(conj), rad where elem(P ) = 1(Z(P )) and rad(E) 2 Sylp(CL(E)). (b)Assume E = 1(Z(P )) and P 2 Sylp(CL(E)). Then the natural map NL(P )=P -- i AutL(E) induced by restriction is a surjection, its kernel has order prime to p, and* * the action of NL(P )=P on Z(P ) factors through AutL(E). Furthermore, for any Aut(L) which contains Inn(L), the natural map N (P )=P -- i Aut (E) is a surjectio* *n. (c)If E 2 Ei(L ; p), then E is a pivotal subgroup. (d)If E is a pivotal subgroup and P 2 Sylp(CL(E)), then P is a radical p-subgro* *up of L. 28 BOB OLIVER (e)Fix S 2 Sylp(L). Each pivotal p-subgroup of L is L-conjugate to a subgroup E* * S such that E = 1(Z(CS(E))), and hence such that E 1(Z(S)). (f)Let E L be an elementary abelian p-subgroup, and let X CL(E)r E be a CL(E)-conjugacy class of elements of order p such that |X| is prime to p. Th* *en no elementary abelian subgroup E0 E such that E0\ X = ? is pivotal in L. Proof.(a) Assume P 2 Ri(L ; p), and let Aut(L) and Z0 Z(P ) be such that i(N (P )=P, NL(P )=P ; Z0) 6= 0. Set H = CL(E) for short. Then P H by definition of E. Since NH (P )=P acts trivially on E, it acts trivially on n(Z0)= n-1(Z0) for each n, and hence must* * have order prime to p by Proposition 3.3(b). So by Lemma 1.11, P 2 Sylp(H). Now assume E 2 Ei(L ; p). By definition, E = 1(Z(P 0)) for some P 02 Ri(L ; * *p), and we just saw that P 02 Sylp(CL(E)). Hence if P 2 Sylp(CL(E)), then P = xP 0x* *-1 for some x 2 CL(E), so P 2 Ri(L ; p), and 1(Z(P )) = xEx-1 = E. (b) Fix Inn(L) Aut (L) (and we identify L with Inn(L) C ). Since E = 1(Z(P )), we have N (P ) N (E). Since P is a Sylow p-subgroup of CL(E) C N (E), conjugation by any x 2 N (E) sends P to hP h-1 for some h 2 CL(E), and so h-1x 2 N (P ). This shows that N (E) = CL(E).N (P ), and hence that N (P )=P surjects onto Aut (E) ~=N (E)=C (E). When = L, then the kernel of this surjec* *tion is NCL(E)(P )=P , which has order prime to p since P 2 Sylp(CL(E)). It remains to show that NCL(E)(P )=P acts trivially on Z(P ). It acts trivia* *lly on E = 1(Z(P )) by definition, and hence on n(Z(P ))= n-1(Z(P )) for each n. By [Gor , Corollary 5.3.3], any group of automorphisms of Z(P ) with this property* * must be a p-group; and since NCL(E)(P )=P has order prime to p, it must act triviall* *y. (c) Assume E 2 Ei(L ; p), and fix P 2 Sylp(CL(E)). Then E = 1(Z(P )) by (a). * *So to show E is pivotal, it remains to show that Op(Aut L(E)) = 1. Let Z0 Z(P ) and Aut(L) be such that i(N (P )=P, NL(P )=P ; Z0) 6= 0. By (b), NL(P )=P surjects onto Aut L(E) with kernel K=P def=NCL(E)(P )=P of ord* *er prime to p, and K acts trivially on Z(P ) (hence on Z0). So by Proposition 3.3(* *d), 0 i N (P )=K, AutL(E); Z 6= 0, and hence Op(Aut L(E)) = 1 by Proposition 3.3(c). (d) Assume E is pivotal and P 2 Sylp(CL(E)). By (b), NL(P )=P surjects onto AutL(E) with kernel of order prime to p. Since Op(Aut L(E)) = 1 by definition * *of a pivotal subgroup, Op(N(P )=P ) = 1, and so P is radical. (e) Fix S 2 Sylp(L). Any pivotal subgroup of L is L-conjugate to a subgroup E such that some P 2 Sylp(CL(E)) is contained in S. Thus P = CS(E), and so E = 1(Z(CS(E))) by definition of pivotal. Also, Z(S).P CS(E) = P , so Z(S) P . Hence Z(S) Z(P ), so that 1(Z(S)) 1(Z(P )) = E. (f) Fix an elementary abelian p-subgroup E L, and a CL(E)-conjugacy class X CL(E) of elements of order p such that |X| is prime to p. Assume E0 E is s* *uch that E0\ X = ?. Then CL(E0) \ X is the fixed point set of an E0=E-action on X, * *hence also has order prime to p, and contains a CL(E0)-conjugacy class X0of order pri* *me to EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 29 p. Fix x 2 X0. Then [CL(E0):CL(E0, x)] is prime to p, so there is P 2 Sylp(CL* *(E0)) such that x 2 Z(P ), 1(Z(P )) E0, and thus E0 is not pivotal. It remains to list some more conditions which can be used later to prove that* * certain subgroups are not in E 2(L ; 2) or R 2(L ; 2). Almost all of these will be base* *d on the following, very general, proposition. Recall (from Section 3) that a radical p-* *chain in a group G is a sequence Op(G) = P0 C P1 C . .C.Pn of distinct p-subgroups of G such that Pi= Op NG(P0, . .,.Pi) for all i (in pa* *rticular, PiC Pn for all i), and such that Pn 2 SylpNG(P0, . .,.Pn-1) . Proposition 4.5. Fix a finite centerfree group L and an elementary abelian p-su* *bgroup E L. Assume, for some m 1, that E 2 Em (L ; p). Then for any sequence 1 = E0 E1 . . . Ek = E of NAut(L)(E)-invariant subgroups, there is some 1 i k, and a radical p-chain 1 = P0 P1 . . . Pm in Aut L(E), such that Ei=Ei-1 (when regarded as a Fp[Pm ]-module) contains a copy of the free mo* *dule Fp[Pm ]. Proof.Fix P 2 Sylp(CL(E)); then P 2 Rm (L ; 2) by Proposition 4.4(a). Let Aut(L) and Z0 Z(P ) be such that m (N (P )=P, NL(P )=P ; Z0) 6= 0. By Proposi* *tion 4.4(b), the natural map NL(P )=P ---! AutL(E) is surjective with kernel of ord* *er prime to p, and the action of NL(P )=P on Z(P ) (hence on Z0) factors through A* *utL(E). Since each quotient n(Z0)= n-1(Z0) is isomorphic as an Fp[N (P )=P ]-module * *to a submodule of E, and since the Eiare all Aut (E)-invariant (hence N (P )=P -inva* *riant) by assumption, there is a sequence 1 = Z0 Z1 . . .Zk = Z0of N (P )=P -invar* *iant subgroups such that each Zi=Zi-1is isomorphic to a subgroup of some Ej=Ej-1. He* *nce it suffices to show that at least one of the quotients Zi=Zi-1 contains a free * *module Fp[Pm ] for some radical p-chain ending in Pm , and this follows from Propositi* *on 3.5. The next proposition gives some special cases of Proposition 4.5. Proposition 4.6. Fix a finite centerfree group L and an elementary abelian 2-su* *bgroup E L. Let 1 = E0 E1 . . .Ek = E be any sequence of NAut(L)(E)-invariant subgroups such that at least one of the following conditions holds for each i: (a)The AutL (E)-action on Ei=Ei-1is not p-faithful; i.e., its kernel has order * *a mul- tiple of p. (b)rk(Ei=Ei-1) 3. (c)rk(Ei=Ei-1) 7 and the Sylow 2-subgroups of Aut L(E) are neither dihedral n* *or semidihedral. (d)rk(Ei=Ei-1) < 2.|R| for each radical 2-subgroup 1 6= R AutL(E). (e)rk(Ei=Ei-1) < |R2| for each sequence 1 6= R1 R2 L such that R1 is a radi* *cal 2-subgroup of L and R2 is a radical 2-subgroup of NL(R1). Then E =2E 2(L ; 2). Proof.Assume otherwise: that E 2 Em (L ; 2) for some m 2. By Proposition 4.5, there is some i, and a radical 2-chain 1 = P0 P1 . . .Pm in AutL(E), such t* *hat 30 BOB OLIVER Ei=Ei-1contains Fp[Pm ] as a summand. In particular, rk(Ei=Ei-1) |Pm | 2m ,* * and this is impossible if any of the conditions (b), (d), or (e) hold. If rk(Ei=Ei-1) 7, then |Pm | 4, so m = 2, P1 = for some involution x,* * and P2 = CS(x) for some S 2 Syl2(Aut L(E)). By [Hp , III.14.23], |CS(x)| = 4 implie* *s that S has maximal class (its central series has length k - 1 if |S| = 2k). By [Gor * *, Theorem 5.4.5], each 2-group of maximal class is dihedral, quaternion, or semidihedral.* * Together with Proposition 3.3(f), this shows that S must be dihedral or semidihedral. It remains to considerfcasei(a). Let H C Aut L(E) be the kernel of the actio* *n on Ei=Ei-1, and assume pfi|H|.fByiLemma 1.11, for any p-subgroup P AutL(E), eith* *er P \ H 2 Sylp(H) or pfi|NHP (P )=P |; in either case N(P ) \ H has order a multi* *ple of p. Upon applying this inductively, we see that N(P1, . .,.Pm-1 ) \ H has order a m* *ultiple of p. Since Pm is assumed to be a Sylow p-subgroup of this intersection of norm* *alizers, this shows that Pm \ H 6= 1, which contradicts the assumption that Zi=Zi-1conta* *ins a summand isomorphic to Fp[Pm ]. We next look at some cases where AutL (E) is a symmetric group or general lin* *ear group. Similar results involving other classical groups in characteristic two * *will be shown in Proposition 6.4. Proposition 4.7. Fix a finite centerfree group L, a pivotal 2-subgroup E L, a* *nd NAut(L)(E)-invariant subgroups 1 = E0 E1 . . .Ek = E. Let Aut L(Ei=Ei-1) denote the image of Aut L(E) in Aut (Ei=Ei-1). Assume, for each 1 i k, that Ei=Ei-1either satisfies one of the conditions (a-e) in Proposition 4.6, or sati* *sfies one of the following conditions: either (a)Aut L(Ei=Ei-1) ~=GLn(2) for some n and rk(Ei=Ei-1) < 2n; or (b)Aut L(Ei=Ei-1) ~= n or An, and Ei=Ei-1 is the permutation representation on (Z=2)n or (Z=2)n-1; or (c) Aut L(Ei=Ei-1), Ei=Ei-1 ~= GLm (2) o n, (Z=2)mn for some m 3 and n 2. Then E =2E 2(L ; 2). Proof.Set E0 = Ei=Ei-1and G = AutL(Ei=Ei-1) for short. By Propositions 4.5 and 4.6, it suffices to show that there is no radical 2-chain 1 P1 . . .Pk of l* *ength k 2 in G such that E0 contains a copy of F2[Pk] as a summand. When G ~=GLn(2), then the smallest radical 2-subgroups of G have order 2n-1 (* *see Lemma 6.1 for a description of the radical 2-subgroups). Hence by Proposition 4* *.6(d), E 2 E 2(L ; 2) implies rk(Ei=Ei-1) 2n. Now assume G ~= n or An, regarded as acting on a set X of n elements, such th* *at Ei=Ei-1 is isomorphic to a submodule or quotient module of F2(X). Then for any 2-subgroup P G, Ei=Ei-1 contains a free submodule F2[P ] only if the action o* *f P on X has at least one free orbit. Furthermore, no radical 2-subgroup P G can * *have m > 1 free orbits: this is clear if |P | = 2, m = 2, and G = An, and holds in t* *he other cases since O2(N n(P )=P ) contains a copy of P m-1. Thus there is no radical 2* *-chain of length 2 such that P2 has a free orbit, and so (b) follows from Propositio* *n 4.5. The third case now follows since for any radical 2-chain {Pi} in GLm (2) o n* * of length 2, either P2 \ (GLm (2))n 6= 1 and contains a nontrivial radical 2-sub* *group of (GLm (2))n (Lemma 1.12), in which case the module contains no summand F2[P2]; or EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 31 P2\ (GLm (2))n = 1, and an argument similar to that used for G ~= n shows that * *the module contains no summand F2[P2]. It is important to note that there is an action of 5 on (Z=2)4 with the prop* *erty that 2( 5; (Z=2)4) ~= Z=2. This is the action obtained by identifying 5 ~= L* *2(4) _ the group SL2(4) ~=A5 extended by its field automorphism _ and (Z=2)4 ~=(F4)2. Note that this action is transitive on the nonzero elements in the module, unli* *ke the permutation action. The following variant of Proposition 4.7 will be needed when handling classic* *al groups in odd characteristic. Proposition 4.8. Fix a finite centerfree group L and a pivotal 2-subgroup E L, and set = Aut L(E) for short. Assume there is a finite set X with -action s* *uch that E is isomorphic to a -submodule of the permutation representation F2(X). * *Let X1, . .,.Xk X be the -orbits, and assume that the -action contains all perm* *utations whose restriction to each Xi is an even permutation. Then E =2E 2(L ; 2). Proof.Assume otherwise. By Proposition 4.5, there is a radical 2-chain 1 P1 * * . . . Pk of length k 2 in G such that X contains a free orbit of Pk. Let Xi X be a -orbit which contains a free Pk-orbit. Let m be the number of free orbits of P1 in Xi (thus m [P2:P1] 2), and let Y Xi be the union of * *those orbits. Let H ~= P1 o m be the group of all permutations of X which centraliz* *e P1 and are the identity on Xr Y . Since contains all even permutations of Xi (he* *nce of Y ) by assumption, \ H C C (P1) has index at most 2 in H. Also, O2(H) ~=P1m if m > 2 and O2(H) = H if m = 2. In either case, O2( \ H) P1, so O2(N (P1)) 6= * *P1, and this contradicts the assumption that P1 is a radical 2-subgroup of . The following proposition will be useful when working with the sporadic group* *s. Proposition 4.9. Let L be any finite centerfree group, let E L be a pivotal p- subgroup, and fix P 2 Sylp(CL(E)). In particular, E = 1(Z(P )). Let H L be* * a subgroup which contains NL(E) NL(P ). (a)Assume Op(H) 6= 1, and set E0 = E \ Z(Op(H)). Then P Op(H), [E, Op(H)] = 1, E0 6= 1, and E0 is an AutL (E)-invariant submodule of E. In particular, i* *n this case, E Z(Op(H)) if E is AutL (E)-irreducible or if Op(H) is centric in H. (b)Assume P 2 Ri(L ; p). Let H0 C H be a characteristic subgroup which is qua- sisimple, and set K = CH (H0). Assume that NAut(L)(E) NAut(L)(H), and that P \ H02=Sylp(H0). Then P K=K 2 Ri(H=K ; p). Proof.(a) Since E is pivotal, P is a radical p-subgroup of L (Proposition 4.4(* *d)), and hence a radical p-subgroup of H (recall NL(P ) = NH (P )). So P Op(H), and E Z(P ) centralizes Op(H). In particular, E0 def=E \ Z(Op(H)) = 1 Z(P ) \ Z(Op(H)) , and so E0 6= 1 since Z(P ) \ Q 6= 1 for any 1 6= Q C P . Also, E0 is AutL(E)-in* *variant, since NL(E) = NH (E) normalizes both E and Op(H). (b) We now assume that P 2 Ri(L ; p) (hence E 2 Ei(L ; p)), that H0 C H is a characteristic subgroup which is quasisimple, that P \ H0 =2Sylp(H0), and that 32 BOB OLIVER NAut(L)(E) NAut(L)(H). Let Aut(L) and Z0 Z(P ) be such that i(N (P )=P, NL(P )=P ; Z0) 6= 0. Set K = CH (H0), and consider the groups Z0 = Z0\ K = Z0\ CH (H0) and Z00= Z0=Z0 ~=Z0K=K H=K. By assumption, N (P ) N (H), and N (H) N (H0) since H0is characteristic in * *H. Hence Z0 is an N (P )-invariant subgroup of Z0. Since P \ H02=Sylp(H0), NPH0(P * *)=P has order a multiple of p by Lemma 1.11. This group acts trivially on Z0 K, a* *nd thus i(N (P )=P, NL(P )=P ; Z00) 6= 0 by Proposition 3.3(b). Now set __ __ Kb = C (H0) and K = CL(H0), so that K = H \ bK and K = L \ bK. The action of NL(P )=P on Z00must_be p-faithful by Proposition 4.6(a). Since P * * H normalizes H0 and hence K and K , NPK_(P )=P acts trivially on Z00= Z0=(Z0\ K) = __ __ Z0=(Z0\ K ). Hence NPK_(P )=P has order prime to p, and P 2 Sylp(P K) by Lemma 1.11 again. Set P 0= P K=K and 0= Aut (H0) ~=N (H0)=C (H0). Recall that K = bK\ H. We can identify P 0 H=K 0 and Z00 Z(P 0), ~=PKb=Kb~=HKb=Kb ~=Z0bK=Kb __ as subgroups of Aut(H0). Since P 2 Sylp(P K) (hence P 2 Sylp(P K)), and N (P ) __ N (H) normalizes K and K , __ __ NH (P K) = NL(P ).K and N (P K) = N (P ).K . We can thus identify NL(P )=P NH=K (P 0)=P 0~=NH (P K)=P K = (NL(P ).K)=P K ~=__________ , NPK (P )=P where NPK (P )=P has order prime to p. Also, since N (P ) N (H) N (H0), __ __ N (P ).bK N (H0) \ N (P bK) N (P K) = N (P ).K N (P ).bK; so these are all equal, and N (P )=P N 0(P 0)=P 0~=NN (H0)(P bK)=P bK= (N (P ).bK)=P bK~=__________ . NPKb(P )=P Hence by Proposition 3.3(d), 0 0 0 0 00 i 00 i N 0(P )=P , NH=K (P )=P ; Z ~= N (P )=P, NL(P )=P ; Z 6= 0, and this proves that P 0= P K=K 2 Ri(H=K ; p). EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 33 5. Alternating groups We are now ready to go through the list of all finite nonabelian simple group* *s L, to show in each case that L 2 L 2(2). When L is of Lie type in characteristic 2* *, this will be done using Proposition 4.3. In all other cases, we prove L 2 L 2(2) wit* *h the help of Proposition 4.2. This means either showing that R 2(L ; 2) = ? (equival* *ently E 2(L ; 2) = ?); or else choosing a 2-centric subgroup Q L which is weakly cl* *osed in a Sylow subgroup of L, and then showing that R 2(L ; 2) Q = ? (equivalently E 2(L ; 2) Z(Q)= ?). We will be constantly referring to Proposition 4.4 for some of the most basic* * prop- erties of subgroups P 2 Ri(L ; 2) and E = 1(Z(P )) 2 Ei(L ; 2), as well as the* * corre- spondence between these two sets. Propositions 4.5, 4.6, 4.7, and 4.8, all of w* *hich give restrictions on the pairs (E, AutL(E)) for E 2 Ei(L ; 2), will then be used to * *further eliminate subgroups from these sets; while Proposition 4.9 will be used in some* * cases to reduce to a problem involving a smaller group. Together with Lemma 1.12, the* *se will be the only results from earlier sections used in this part of the proof. The easiest case to consider is that of the alternating groups. Theorem 5.1. For any n 5, An 2 L 2(2). Proof.Set L = An. When n 7, then rk2(L) = 2, so R 2(L ; 2) = ? by Proposition 4.6(b), and L 2 L 2(2) by Proposition 4.2. So we can assume that n 8, and hen* *ce that Aut(L) = n (cf. [Sz, 3.2.17]). We regard n as the group of permutations of the set n = {1, . .,.n}. For eac* *h i 1, let E2ibe the set of subgroups E 2i, such that E ~=(C2)i, and the action of * *E on n contains one free orbit and is otherwise fixed. These subgroups (for each i) * *clearly make up one conjugacy class in n. Fix some S 2 Syl2(L), and let Q S be the subgroup generated by all E S su* *ch that E 2 E4. For any E, E02 E4 with overlapping support, either E = E0(if they * *have the same support), or is not a 2-group (hence not contained in S). Thus * *Q is the direct product of [n=4] subgroups in E4 with disjoint support, and is weakl* *y closed in S with respect to n by construction. By Proposition 4.2, it remains only to* * show that R 2(An ; 2) Q = ?. Fix P 2 R 2(An ; 2) Q , and set E = 1(Z(P )). Then P 2 Syl2(CAn(E)) by Lemma 4.4(a). Each union of k orbits of E of order q (orbits under the action on n) w* *hich are isomorphic as E-orbits contributes a factor Eq o k (for Eq 2 Eq) to C n(E). He* *nce P is the intersection with An of a product of subgroups of the form Eq o C2 o . .* *o.C2 for Eq 2 Eq. If N(P ) does not act transitively on n, then E is reducible as a N(P* * )=P - module. By Proposition 4.6(a), N(P )=P must act p-faithfully on at least one of* * the n-irreducible composition factors of E, and this implies that all but one of t* *he factors of N(P )=P must have odd order. Hence P must have the form i xr j P = An \ Eq_o_C2_o-.z.o.C2_"x S0 , k times where q 2, Eq 2 Eq, k 0, and S0 2 Syl2( n-q2kr). Since P does not contain a* *ny subgroup conjugate to Q, we must have either q 8, or q = 2 and k = 0. Set E0 = E \ S0. Then Aut L(E) acts trivially on E0, and (Aut L(E), E=E0) is isomorphic to one of the pairs (GLs(2) o r, (Z=2)rs) (if q = 2s, s 3), or ( * *r, (Z=2)r) 34 BOB OLIVER or ( r, (Z=2)r-1) (if q = 2). In either case, Proposition 4.7(b,c) applies to s* *how that E =2E 2(An ; 2) (and hence P =2R 2(An ; 2)). By extending the above arguments, one can show that R 3(An ; 2) = ? for all n, and n x2 fi * * o R2(An ; 2) = E4_o_C2_o-.z.o.C2_"x S0fifiE4 2 E4, 2k+3 n, S02 Syl2(An-2k+3* *) . k times 6. Groups of Lie type in characteristic two We next consider simple groups of Lie type in characteristic two. We first su* *mmarize the structures in these groups which will be needed (in arbitrary characteristi* *c p). We refer to [Ca1 ] and [GLS3 ] as general references. Assume first that L is a Chevalley group: L = G(q), where G is one of the gro* *ups An, Bn, etc., defined over the finite field Fq (q = pa). For example, An(q) ~=P SLn* *+1(Fq). Let V denote the root system of G, where V is a real vector space with inner product. Let + be the set of positive roots; thus = { r | r 2 +}. Let I den* *ote the set of primitive roots, an R-basis of V. Then rk(G) def=dim(V) = |I|. To each root r 2 corresponds a root subgroup Xr ~=Fq in L = G(q), and U def= Q def r2 +Xr is a Sylow p-subgroup of L. Also, B = NL(U) = Uo H (the Borel subgrou* *p), where H (Fq)rk(G)(of index dividing q- 1) is the subgroup of diagonal element* *s. Set N = NL(H); then W ~=N=H is the Weyl group of G (and of the root system ). For example, when L = An(q) = P SLn+1(q), then we can take X V = {x 2 Rn+1| xi= 0}, = {ffli- fflj| i 6= j} (where {ffl1, . .,.ffln+1} is the standard basis of Rn+1), + = {ffli- fflj| i < j}, and I = {ffli- ffli+1}. Then Xffli-ffljis the subgroup of matrices which have 1's along the diagonal an* *d are zero elsewhere except at entry (i, j), and hence U is the group of upper triang* *ular matrices with 1's along the diagonal, and B is the group of all upper triangula* *r matrices. Diagonal elements are those represented by diagonal matrices, N is the image of* * the subgroup of monomial matrices, and W ~= n+1. For any subset J I, let be the set of positive linear combinations of e* *lements of J contained in +. Set fi ff fi ff PJ = U.H, X-r fir 2 , and UJ = Xrfir 2 +r . The PJ are the parabolic subgroups of G(q). (We use this instead of the usual n* *otation PJ to avoid confusion with our use of P for an arbitrary p-group.) Also, UJ = O* *p(PJ), and PJ = NL(UJ) (see [GLS3 , Theorem 2.6.5(d,e)]). We next fix the notation for the twisted groups tG(q). The index t is the ord* *er of an automorphism oe 2 Aut (G(q0)), which is the composite of a graph automorphism induced by ø 2 Aut(V, , +) with a field automorphism (induced by an automorph* *ism of Fq0). In most cases, q0 = qt, and so Fq is the fixed subfield of the automo* *rphism of order t on Fq0. In all cases, ø 2 Aut( +) can be seen as an automorphism of* * the EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 35 Dynkin diagram (and t = 2, 3). Also, oe(Xr) = Xfi(r)for each r 2 , and thus oe* * leaves invariant the subgroups U, H, and N. The twisted group L = tG(q) is defined to * *be 0 the subgroup Op (CG(q0)(oe)) generated by the Sylow p-subgroups of CG(q0)(oe) (* *recall p = char(Fq)), or alternatively as the subgroup of G(q0) generated by U1 def=CU* *(oe) and the analogous subgroup for the root groups of negative roots. Its Borel subgro* *up is defined to be B1 = NL(U1) = U1o H1 where H1 def=CH (oe) \ L. Its parabolic subgroups are the groups P1Jdef=L \ CPJ(oe), for all ø-invariant * *subsets J I. In this case, we again have U1Jdef=CUJ(oe) = Op(P1J), and P1J= NL(U1J) (* *again by [GLS3 , Theorem 2.6.5(d,e)]). 0 Lemma 6.1. Let L = Op (CG(q)(oe)) be group of Lie type in characteristic p (pos* *sibly oe = Id and L = G(q)). Fix a root system (V, , +) for G, let I + be the se* *t of primitive roots, and let ø denote the action on induced by oe. Let U be the p* *roduct of the root subgroups Xr for r 2 +, and set S = U1 2 Sylp(L). Then a subgroup P * * S is radical in L if and only if P = U1Jfor some ø-invariant subset J I. Furthe* *rmore, for each such J, U1Jis weakly closed in S with respect to L, and more generally* * with respect to any subgroup Aut(L) whose action on the set of primitive roots l* *eaves J invariant. Proof.By [BW ], each radical p-subgroup of L is L-conjugate to some U1J. More * *pre- cisely, this is shown in [BW ] for radical p-subgroups of Inndiag(L): the exte* *nsion of L by its diagonal automorphisms. But since the radical p-subgroups of L are preci* *sely the intersections of L with radical p-subgroups of Inndiag(L) (Lemma 1.12(b)), * *the same result holds for radical p-subgroups of L. It remains to show, for any Aut(L) containing Inn(L) and any -invariant * *subset J I, that U1Jis weakly closed in S = U1 with respect to . The following argu* *ment is based on the proof of [Gro , Lemma 4.2]. Let B = P1?= NL(S) be the Borel subgro* *up, and let N be the normalizer of the group of diagonal elements in L. Then L = BNB by, e.g., [Ca1 , 8.2.2]. Hence, since U1JC B, if U1J6= gU1Jg-1 S for any g 2 * *L, then this occurs for g 2 N. Now, g permutes the root subgroups via the action of w = gH 2 W fion , and so w( +r ) +. Write = +r for short; this is closed in the sense that * *any r 2 which is a positive linear combination of elements of also lies in . So w( ) = +r w( ) = +r w(R) has the same property, where R denotes the R-linear subspace of V generated * *by J. Hence w( ) is generated by the set J0 I of primitive roots which it contai* *ns, and thus w( ) = +r . After replacing w by its product with some element in the* * Weyl group of , we can assume that it sends positive roots to positive roots, a* *nd hence must be the identity. But this contradicts the assumption that U1J6= gU1Jg-1. We next list, for future reference, the notation used here for the root syste* *ms of the simple Lie groups. Notation 6.2. The following notation will be used for the root systems of the s* *imple Lie groups. When listed, H V denotes a half-space which contains the positive* * roots, and {ffl1, . .,.ffln} denotes the canonical basis of Rn. 36 BOB OLIVER P P o G = An : V = {x 2 Rn+1| xi= 0}, H = {x | (n - i)xi> 0}, + = {ffli- fflj| i < j}, and I = {ffl1- ffl2, ffl2 - ffl3, . .,.f* *fln- ffln+1}. P o G = Bn : V = Rn, H = {x | (n + 1 - i)xi> 0}, + = {ffli, ffli fflj| i < j}, and I = {ffl1- ffl2, ffl2 - ffl3, . .* *,.ffln-1- ffln, ffln}. P o G = Cn : V = Rn, H = {x | (n + 1 - i)xi> 0}, + = {2ffli, ffli fflj| i < j}, and I = {ffl1- ffl2, ffl2 - ffl3, . .* *,.ffln-1- ffln, 2ffln}. P o G = Dn : V = Rn, H = {x | (n + 1 - i)xi> 0}, + = {ffli fflj| i < j}, and I = {ffl1- ffl2, ffl2 - ffl3, . .,.ffln-* *1- ffln, ffln-1+ ffln}. P o G = G2 : V = {x 2 R3| xi= 0}, H = {(x1, x2, x3) | 2x1 + x2 + 4x3 > 0}, = (ffli- fflj), (2ffli- fflj- fflk) | {i, j, k} = {1, 2, 3} I = {(1, -1, 0), (-2, 1, 1)}. o G = F4 : V = R4, H = {x | 7x1 + 3x2 + 2x3 + x4 > 0}, + = { +(B4), 1_2(ffl1 ffl2 ffl3 ffl4)}, and I = {ffl2- ffl3, ffl3- ffl4* *, ffl4, 1_2(ffl1- ffl2- ffl3- ffl4)}. o G = E8 : V = R8, n fi o n X7 fiX * * o + = fflj fflififi1 i< j 8 [ 1_2 (-1) (i)ffli+ ffl8fifi (i) ev* *en i=1 I = 1_2(ffl1- ffl2- ffl3- ffl4- ffl5- ffl6- ffl7+ ffl8), ffl2- ffl1, ffl2+ f* *fl1, ffl3- ffl2, ffl4- ffl3, ffl5- ffl4, ffl6- ffl5,* * ffl7- ffl6 . o G = E7 : V = {x 2 R8| x7 + x8 = 0}, + is the set of positive roots of E8 which lie in V, and I = 1_2(ffl1- ffl2- ffl3- ffl4- ffl5- ffl6- ffl7+ ffl8), ffl2- ffl1, ffl2+* * ffl1, ffl3- ffl2, ffl4- ffl3, ffl5- ffl4, ffl6- ffl5 . o G = E6 : V = {x 2 R8| x6 = x7 = -x8}, + is the set of positive roots of* * E8 which lie in V, and I = 1_2(ffl1- ffl2- ffl3- ffl4- ffl5- ffl6- ffl7+ ffl8), ffl2- ffl1, ff* *l2+ ffl1, ffl3- ffl2, ffl4- ffl3, ffl5- ffl4 . To see that these do, in fact, describe root systems of the given types, we r* *efer to Bourbaki [Bb , xVI.4]. Theorem 6.3. Let L be a finite simple group of Lie type in characteristic two, * *or the Tits group 2F4(2)0. Then L 2 L 2(2), and L 2 L 1(2) if L 6~=P SL3(2). 0 Proof.Write L C G = O2 (CG(q)(oe)), where q is a power of 2 (and possibly oe = * *Id). Thus L = G, except when L is the Tits group (in which case [G:L] = 2). Fix a ro* *ot system = + [ - for G, let I + be the set of primitive roots, and let ø d* *enote the action on induced by oe. Set S = U1 = CU(oe) 2 Sylp(G), where U is the pr* *oduct of the root subgroups Xr for r 2 +. By Lemma 6.1, a subgroup P S is radical in G if and only if P = U1Jfor some ø-invariant subset J I, each such subgrou* *p is weakly closed in S with respect to G, and is weakly closed with respect to Aut(* *G) if J is invariant under all permutations of I induced by graph automorphisms. Let Z + and J0 I be the subsets defined according to the following tabl* *e: EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 37 _________________________________________________________________ || G | Z | J0 || ||__________|_______________________|____________________________|| || An (n 1)| {ffl1 - ffln+1} | {ffl1 - ffl2, ffln - ffln+1} * * || ||__________|_______________________|___________________________ * * || || Bn (n 2)| {ffl1 + ffl2, ffl1}| {ffl2 - ffl3, ffl1 - ffl2} * * || ||__________|_______________________|___________________________ * * || || Cn (n 3)| {2ffl1, ffl1 + ffl2}| {ffl1 - ffl2, ffl2 - ffl3} * * || ||__________|_______________________|___________________________ * * || || Dn (n 4)| {ffl1 + ffl2} | {ffl2 - ffl3} || ||__________|_______________________|___________________________ || || G2 | {(-1, -1, 2)} | {(-2, 1, 1)} || ||__________|_______________________|___________________________ || || F4 | {ffl1 + ffl2, ffl1}|{ffl2 - ffl3, 1_(ffl1 - ffl2 - ff* *l3 - ffl4)} || ||__________|_______________________|_________________2_________ * * || || E6 | {1_(1, 1, 1, 1, 1, -1,|-1, 1)} {ffl2 + ffl1} || ||__________|__2____________________|___________________________ || || E7 | ffl8 - ffl7 | {1_(ffl1 - ffl2 - . .-.ffl7 + ffl8* *)} || ||__________|_______________________|___2_______________________ * * || || E8 | ffl7 + ffl8 | {ffl7 - ffl6} || ||__________|_______________________|____________________________ || Then by [GLS3 , Theorem 3.3.1], Y Z(U) = Xr. (1) r2 Z Furthermore, by the same reference, when L = 2An(2k), 2Dn(2k), 3D4(2k), or 2E6(* *2k), then Z(S) = Z(U1) = CZ(U)(oe). (2) Both of these are also consequences of Chevalley's commutator formula, to be di* *scussed below. Case 1: We consider here the cases (i)L = G(2k) for some G and some k 2; or (ii)L is one of the groups 2An(2k), 2Dn(2k), 3D4(2k), or 2E6(2k) for k 2; or (iii)L is a Suzuki group 2B2(22k+1) or a Ree group 2F4(22k+1) for some k 1. By [Ca1 , Theorem 6.3.1], for any r 2 , is the image of a homomorp* *hism OEr defined on SL2(q), which sends (strict) upper and lower triangular matrices* * to the root subgroups Xr and X-r, respectively. Let D SL2(q) be the subgroup of diag* *onal matrices. The images OEr(D) commute with each other (for all r 2 ), OEr(D) nor* *malizes Xs for all r and s, the OEr(D) generate the abelian subgroup H of diagonal elem* *ents of G(q), and NG(q)(U) = UH (cf. [Ca1 , x7.1]). In particular, when q = 2k for k* * 2, CXr(OEr(D)) = 1. Thus when L = G(2k) for k 2, then H acts on U with trivial fixed subgroup, * *and hence Z(U)NL(U)= Z(NL(U)) = 1. If L is one of the groups 2An(2k), 2Dn(2k), 3D4(* *2k), or 2E6(2k) for k 2, then Z = {s} has order 1, Z(U1) = CXs(oe) ~=F2k, OEs(SL2* *(2k)) is contained in L and contains CXs(oe), and hence Z(NL(U1)) = 1. Similarly, if* * L is a Suzuki group 2B2(22k+1) or a Ree group 2F4(22k+1) for some k 1, then the ce* *nter of the Borel subgroup is trivial: this follows from the description (cf. [Ca1 ,* * Theorem 13.7.4]) of the diagonal elements in these groups; or (more explicitly) from [H* *B3 , xXI.3] for the Suzuki groups and from [Sh , xII] for the Ree groups. 38 BOB OLIVER Since NL(U1J) NL(U1) for each ø-invariant subset J I, this shows that in * *all of the cases (i), (ii), (iii) above, Z(NL(P ))p = 1 for all p-centric radical p-subgroups P S. (3) So in these cases, the theorem follows from Proposition 4.3, applied with Q = S. Case 2: The groups 2B2(2) ~= C5o C4 and 2A2(2) = P SU3(2) are solvable, and none of the groups B2(2) ~=C2(2) ~= 6, or G2(2) ~=Aut (P SU3(3)) is simple. When L = A2(2) = P SL3(2), then rk2(L) = 2 (its Sylow 2-subgroups are dihedral of or* *der 8), so R 2(L ; 2) = ? by Proposition 4.6(b), and L 2 L 2(2) by Proposition 4.2. So among the simple groups of Lie type, it remains only to consider those whe* *re L = G(2) with rk(G) 3, or where L is one of the groups 2An(2) (n 3), 2Dn(2), 3D4(2), or 2E6(2). In all of these cases, we claim (with reference to the abov* *e table) that (b)J0 is invariant under any permutation of I induced by an automorphism of the root system; (c)U1J0is 2-centric in L; and (d)for each J I, Z(P1J) = Z(P1J\J0). Once we have proven these points, then the theorem follows from Proposition 4.3* * again, this time applied with Q = U1J0. Point (b) is clear in all cases. To see the other two points, we refer to Che* *valley's commutator formula (cf. [Ca1 , Theorem 5.2.2]). This formula describes, for any* * r, s 2 +, the commutator of a pair of elements in Xr and Xs as a product of elements * *in the root subgroups Xir+jsfor all i, j 1 such that ir + js 2 . Upon examining more closely the coefficients in the formula (this is done explicitly in [GLS3 * *, Theorem 1.12.1(b)]), we see that [Xr, Xs] = 1 () r + s =2 +; or G 2 {Bn, Cn, F4}, r, s short roots, r? s, r+ s a long(ro* *ot4) [Xr, Xs] 6= 1, 1 6= x 2 Xr, 1 6= y 2 Xs =) [x, y] 6= 1. When L = G(2) (and rk(G) 3), points (c) and (d) now follow easily from (4). To see (c), since = J0 in all cases, it suffices to check that for each r * *2 J0 there is s 2 +r J0 such that [Xr, Xs] 6= 1, and this is clear. To see (d),Qit suffi* *ces to checkQthat the actions on Z(S) (as described above) by the groups r2X-r and r2X-r have the same fixed subgroup. More precisely, in all cases, ther* *e is a map of sets ! : J0 ___! Z with the following property: Z(PJ) is the product o* *f the root subgroups Xr for r 2 !(J \ J0). This is easily checked for the classical g* *roups and F4(2). In the remaining cases E6(2), E7(2), and E8(2), the argument is simplifi* *ed by using the descriptions in [Bb , pp.250-270] of the positive roots as linear com* *binations of primitive roots. In all of the remaining (twisted) cases, Z = {s} has just one element, hence* * Z(S) = Xs1, and so (d) can be checked using directly the formulas in (4) again. Simil* *arly, |J0| = 1 (and point (c) is easily checked), except when L = 2An(2) ~=P SUn+1(2)* * for some n 3. In this case, if we take r = ffl1- ffl2 2 J0 and s = ffl2- ffln (so* * ø(s) = s), then by (4), [Xr, Xfis] = 1 and [Xfir, Xs] = 1, and hence no nontrivial element of C* *Xr.Xør(oe) centralizes CXs.Xøs(oe) U1J0. Thus U1J0is 2-centric in S, and (c) holds in th* *is case. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 39 Case 3: Now assume L is the Tits group 2F4(2)0. By Lemma 1.12(b), the radical 2-subgroups of L are precisely the subgroups L \ P for 2-radical subgroups P * *2F4(2). Hence by Lemma 6.1, the subgroups of L \ CU(oe) 2 Syl2(L) which are radical in * *L are exactly the subgroups L \ CUJ(oe) for ø-invariant subsets J I. Since I has ju* *st two ø-orbits, this means that there are exactly two such proper subgroups. By [P1 ] (and in the notation used there), there are 2-subgroups J, K L, wi* *th normalizers H = NL(J) and N = NL(K), such that H and N both contain T 2 Syl2(L), and O2(H) = J, H=J ~=C5o C4, O2(N) = K, N=K ~= 3. Thus J and K are the two radical subgroups of L which are proper subgroups of T* * . Since |J| 6= |K|, this implies that J and K are both weakly closed in T with re* *spect to Aut (L). Also, by [P1 ] again, Z(H) = Z(T ) has order 2 (H is chosen to be * *the centralizer of an involution), while Z(K) ~=C22. Hence L 2 L 1(2) by Propositio* *n 4.3, applied with Q = K. Some more precise results will be needed when working with some of the other * *simple groups. Proposition 6.4. Fix a finite centerfree group L, a pivotal 2-subgroup E L, a* *nd P 2 Sylp(CL(E)). Fix NAut(L)(E)-invariant subgroups 1 = E0 E1 . . .Ek = E. Assume, for some 1 j k, that Ei=Ei-1 satisfies one of the conditions (a-e) * *in Proposition 4.6 for each i 6= j, and that one of the following conditions holds* * for Ej=Ej-1: either (a)(Aut L(Ej=Ej-1), Ej=Ej-1) ~=(Sp2n(2); (Z=2)2n) for n 2; or (b)(Aut L(Ej=Ej-1), Ej=Ej-1) ~=( n(2); (Z=2)n) for n 5; or (c)(Aut L(Ej=Ej-1), Ej=Ej-1) ~=(SO2n(2); (Z=2)2n) for n 3; or (d)(Aut L(Ej=Ej-1), Ej=Ej-1) ~=(G2(2), (Z=2)6). Here, AutL(Ej=Ej-1) is the image of AutL(E) in Aut(Ej=Ej-1). Then E =2E 2(L ; 2* *). Proof.This is closely related to a theorem of Grodal [Gro , Theorem 4.1], but d* *oes not seem to follow from that result (at least not easily) in the generality we need* * it here. Set E0= Ej=Ej-1 and G = AutL(Ej=Ej-1) for short. By Proposition 4.5, it suffi* *ces to show that there is no radical 2-chain 1 P1 . . .Pk of length k 2 in G * *such that E0 contains a copy of F2[Pk] as a summand. When G ~=G2(2), the smallest nontrivial radical 2-subgroups of G are those UJ* * for |J| = 1, and have order 25. When G ~=Bn(2) ~= 2n+1(2) or Cn(2) ~=Sp2n(2) (for n 2), then using Lemma 6.1 and the descriptions of the root systems above, we see that the smallest no* *ntrivial radical subgroup of G is UJ for J = {ffl2 - ffl3, ffl3 - ffl4, . .,.ffln-1- ffln, ffln} or J = {ffl2 - ffl3,* * ffl3 - ffl4, . .,.ffln-1- ffln, 2ffln}, respectively. These correspond to the cases PJ=UJ ~= Bn-1(2) or Cn-1(2). In b* *oth cases, UJ is the product of the root groups for positive roots ffl1 ffli(2 i* * n), and ffl1 or 2ffl1. So |UJ| = 22n-1. Hence the second term of any radical 2-chain in G ha* *s order at least 22n > 2n + 1. 40 BOB OLIVER Now assume G ~= 2n(2) for n 3. When n 4, the smallest radical subgroup occurs for J = {ffl2- ffl3, ffl3 - ffl4, . .,.ffln-1- ffln, ffln-1+ ffl* *n}, UJ is the product of root groups for roots ffl1 ffli, and hence has order |22* *n-2|. Since 22n-1> 2n for n 3, the result again follows from Proposition 4.5. When n = 3,* * the smallest radical subgroups have order 23 (for G ~= +6(2)) or 24 (for G ~= -6(2)* *); and both of these have order > 6. Now assume G ~=SO2n(2) (n 3), and set G0= [G, G] ~= 2n(2). For any radical 2-subgroup 1 6= P G, P \ G0 is a radical 2-subgroup of G0 by Lemma 1.12(b). Then either P \ G0 6= 1, in which case |P | 22n-2 (or |P | 23 if n = 3); or |P | = 2 and P \ 2n(2) = 1. In this last case, one can show by directly study* *ing centralizers of involutions in SO2n(2) that P must be generated by an orthogon* *al transvection: an involution which fixes a codimension one subspace of F2n2, and* * such that NG(P ) = CG(P ) ~=C2xSp2n(2). Thus in either case, the second term of a ra* *dical 2-chain has order at least 22n-2> 2n. We finish the section with the following, stronger result, which involves one* * special case of Theorem 6.3 which will be needed when handling some of the sporadic gro* *ups. Lemma 6.5. Set L = P SU6(2). Then for any Aut (L) containing Inn(L), R 2( ; 2) = ?. Proof.A general description of the outer automorphism group of a finite simple * *group of Lie type is given in [GLS3 , Theorem 2.5.12]. In the notation of that theore* *m, when L = P SU6(2) = 2A5(2), then Outdiag(L) ~=C3, L ~=C2 and acts on Outdiag(L) via x 7! x2, and L = 1. Thus Out(L) ~= 3. Fix P 2 R 2( ; 2), and set E = 1(Z(P )) 2 E 2( ; 2). In particular, P is a* * radical 2-subgroup of (Proposition 4.4(c,d)), and so P0 def=P \ L is a radical 2-subg* *roup of L by Lemma 1.12(b). Also, P0 6= 1, since P 2 Sylp(C (E)) is 2-centric. Thus NL(P0* *) L is a proper parabolic subgroup. There are three parabolic subgroups H L for which the corresponding radical subgroup O2(H) has center of rank 4: NL(P1) = H1 ~=29:L3(4), NL(P2) = H2 ~=24+8:( 3 x A5), NL(P3) = H3 = H1 \ H2 ~=24+5+4:A5. These correspond to the sets of primitive roots J1 = {ffli- ffli+1| i = 1, 2, 4* *, 5}, J2 = {ffli- ffli+1| i = 1, 3, 5}, and J3 = {ffli- ffli+1| i = 1, 5}, respectively. T* *heir normalizers in Aut(L) have the form NAut(L)(P1) = NAut(L)(H1) ~=29:L3(4): 3, NAut(L)(P2) = NAut(L)(H2) ~=24+8:( 3 x L2(4)), NAut(L)(P3) = NAut(L)(H3) ~=24+5+4: L2(4). Here, L2(4) ~=(3xA5):2 is the extension of GL2(4) by the field automorphism. T* *hese are the only possibilities when P = P0 L; if P L then there is the addition* *al possibility NAut(L)(P4) ~=26+4:L3(2). EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 41 Since the smallest nontrivial radical subgroups of L3(4) have order 16, Propo* *sition 4.6(d,e) shows that P1 =2R 2( ; 2) for any choice of . In all of the remainin* *g cases, rk(E) = rk(Z(P )) 7, and in almost all of the cases Aut (E) (for any choice o* *f ) contains no radical p-chain of length 2 ending with a subgroup of order 4 (so E* * 2= E 2( ; 2) by Proposition 4.5). The only exceptions to this occur when P = P2 or P3 and =L has even order. In these cases, Aut (E) ~= 5. Identify GU6(2) GL6(4) as the subgroup of matrices* * A such that Ab= A-1, where A 7! Abis the äb ckward transposition" (aij$ a7-j,7-i) followed by the field automorphism.ijIn either case (P = P2 or P3), E is the gr* *oup I0X of matrices of the form 0I 0 , where each entry is a 2 x 2 block and X = Xb. 00 I Then E contains two different conjugacy classes of involutions _ five transvect* *ions (5-dimensional fixed subspace) and ten involutions with 4-dimensional fixed sub* *space _ and this is possible only if 5 acts via the permutation action. So by Propos* *ition 4.7(b), P2 and P3 cannot lie in R 2( ; 2). 7. Classical groups of Lie type in odd characteristic We next show that when q is an odd prime power, the simple classical groups P SLn(q), P SUn(q), P Sp2n(q), and P n(q) are all in L 2(2). We refer to [Di]* * or [Ca1 , x1] for definitions and descriptions of these groups. Recall that the modular character ØV of an Fq[G]-module V is defined by ident* *ifying F*qwith a subgroup of C*, and then letting ØV (g) 2 C be the sum of the eigenva* *lues g of V ___! V lifted to C. We always consider this in the case where G has order* * prime to q, and hence when two representations with the same character are isomorphic* *. See [Se, x18] for more details. Lemma 7.1. Assume G = Aut(V, b), where V is a finite dimensional vector space o* *ver a finite field K of odd characteristic p, and b is a symplectic, quadratic, her* *mitian, or trivial form. Fix H G of order prime to p, let Ø: H ___! C be the character * *of V as an H-representation, and let Autffl(H) be the group of automorphisms ff 2 Au* *t(H) such that Ø Off = Ø. Then (a)Aut ffl(H) = AutG (H) if G is a linear or unitary group; (b)Aut ffl(H) = AutG (H) if there is no irreducible H-representation W V such* * that b|W 6= 0; and (c)[Aut ffl(H): AutG(H)] 2 if G is an orthogonal or symplectic group and V ~=* * W k for some irreducible H-representation W . Proof.We give here a direct proof; this can also be shown as a consequence of P* *ropo- sition 8.5 in the next section. Clearly, Aut G(H) Aut ffl(H). If H Aut (V ) has order prime to p, and ff* * 2 Autffl(H), then the two representations of H on V induced by the inclusion H * *Aut(V ) and by its composite with ff are isomorphic. Thus Autffl(H) AutAut(V()H). We now study how this can be done while preserving a hermitian, symplectic, or symmetric form b. For any irreducible summand W V , either b|W is nonsingul* *ar 42 BOB OLIVER (hence V = W W ?), or b|W = 0 and V splits as a direct sum of some W W 0(whe* *re b defines an isomorphism W 0~=W *) with its orthogonal complement. We can thus decompose V as orthogonal direct sums _ k ! M M` V = Wi V 0 where V 0= Uj U0j, (1) i=1 j=1 where each Wi, Uj, and U0jis an irreducible H-representation, all irreducible s* *ubrep- resentations of V 0are isotropic (b vanishes on them), and b defines an isomorp* *hism Uj ~=(U0j)*. These conditions determine V 0uniquely, and determine the Wi, Uj, * *and U0jup to isomorphism. Hence for any ff 2 Autffl(H), there is an ff-linear autom* *orphism ' 2 Aut(V ) which permutes these summands. Also, ' can be chosen to preserve b * *on each Uj U0j, since otherwise we could compose it by an appropriate H-linear e* *ndo- morphism of Uj (and the identity on U0j). This shows that ff 2 AutG (H) if V = * *V 0; i.e., if b|W = 0 for all irreducible W V . Next assume that b is a symplectic or symmetric form, and that V ~= W kfor so* *me irreducible H-representation W . We must show that [Aut ffl(H): AutG(H)] 2. B* *y the decomposition (1), it suffices to do this when V is the orthogonal direct sum o* *f copies of W ; in particular when W supports a form of the same type as b. From this, w* *e are easily reduced to the case where k = 1, and hence V = W is irreducible. Set bK= EndK[H](V ), a field. Let K0 bKbe the subfield of all elements r 2 b* *Ksuch that b(rx, y) = b(x, ry) for x, y 2 V . For any automorphism ' 2 NAut(V()H), th* *ere is a unique K-linear automorphism oe = !(') of V such that b('(x), '(y)) = b(x, oe(y)) for all x, y 2 V ; and one easily sees that oe is H-linear and hence oe 2 Kb*. * * By the (anti-)symmetry of b, b(oe(x), y) = b(x, oe(y)) for all x, y, and thus oe 2 (K0* *)*. This defines a homomorphism ! : NAut(V()H) -----! (K0)*, where !(u.Id) = u2 for all u 2 (K0)*. The subgroup T def=!(CAut(V()H)) thus con* *tains all squares in (K0)*, and so 0 * Aut G(H) = Ker NAut(V()H)=CAut(V()H) ---! (K ) =T has index at most two in NAut(V()H)=CAut(V()H) ~=Autffl(H). It remains to consider the case where b is a hermitian form; we claim that Au* *tG(H) = Autffl(H). Let K0 K be the subfield of index 2: the fixed field of the invo* *lution r 7! ~rused to define b. Using the decomposition (1), it suffices to show, for* * any irreducible K[H]-representation W which supports a hermitian form, and any two * *H- invariant forms b, b0on W , that there is an K[H]-linear map ': W ---! W such * *that b0(x, y) = b('(x), '(y)) for all x, y 2 W . Let oe 2 End K[H](W ) be such that b0(x, y) = b(x, oe(y)). Since both forms * *are hermitian, b(x, oe(y)) = b(oe(x), y) for all x and y. Let Kb0 End K[H](W ) b* *e the subfield generated by oe and K0, and let bKbe the subfield generated by oe and * *K. We regard W as a bK[H]-module. Then b(x, ry) = b(rx, y) for all r 2 bK0, so bK0\ K* * = K0. Thus [Kb0:K0] = [Kb:K] is odd, and [Kb:Kb0] = 2. Let r 7! ~rbe the automorphism* * of bK of order 2. Then b(rx, y) = b(x, ~ry) for all x, y and all r 2 bK. So if we cho* *ose r such that r~r= oe, then b0(x, y) = b(rx, ry), and this finishes the proof. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 43 The next lemma will be needed later as an explicit way of constructing automo* *r- phisms. Lemma 7.2. Assume G = Aut(V, b), where V is a finite dimensional vector space o* *ver a finite field K of odd characteristic p, and b is a symplectic, quadratic, her* *mitian, or trivial form. Fix H G of order prime to p, and let T Z(H) be an elementary abelian 2-subgroup. Set fi = ff 2 AutG (H) fiff|T = Id, ff Id (mod { Id}) . For ff 2 and x 2 T , define ffx 2 Aut (H) by setting ffx(g) = g if ff(g) = g * *and ffx(g) = xg otherwise. Then ffx 2 AutG (H) for all ff 2 and x 2 T . Proof.Let T Z(H) and AutG (H) be as described. Fix x 2 T and ff 2 , and let V be the 1-eigenspaces for x. Then V = V+ V- is an orthogonal direct su* *m of H-invariant subspaces. By assumption, there is an ff-linear automorphism ' 2 Aut(V, b), and ' preser* *ves the V since ff(x) = x. Define _ 2 Aut(V, b) by setting _|V+ = Id and _|V- = '|* *V-. Then _ is ffx-linear, and hence ffx 2 AutG (H). Some more information is needed about elementary abelian 2-subgroups of the p* *ro- jective classical groups. Lemma 7.3. Assume G = Aut (V, b), where V is a finite dimensional vector space over a finite field K of odd characteristic, and b is a symplectic, quadratic, * *hermitian, or trivial form. Set Z = Z(G), and let eE G be a subgroup containing Z such th* *at eE=Z is an elementary abelian 2-group. Set eE0= Z. 1(Z(Ee)). Let Ø: eE___! C b* *e the character of V as an eE-representation. Then (a)Ø(g) = 0 for all g 2 eEreE0; and fi (b) def=ff 2 Aut(Ee) fiff|Ee0= Id, ff Id (mod { Id}) AutG (Ee). Proof.Write Ee ~=T x Z0xC2 X , where T is elementary abelian, X is an extraspecial 2-group (or X ~=C2 if eEis * *abelian), and Z0 is a cyclic group with [Z0:Z] 2. Under this identification, Z(Ee) = T* * x Z0 and Ee0= T x Z. For any g 2 Ee, we let -g denote g.(- Id). Since V splits as* * an orthogonal direct sum indexed by the characters of T , we are immediately reduc* *ed to the case where T = 1; i.e., when eE= Z0xC2 X and eE0= Z. Point (a) follows from (b) since Ø(g) = 0 whenever there is ff 2 Aut G(Ee) su* *ch that ff(g) = -g. For any ff 2 , ff 2 Hom G (Ee) if ff|Z0 = Id since it is an * *inner automorphism of eE. This proves (a) and (b) when Z0= Z. Now assume [Z0:Z] = 2. By the above remarks, Ø(g) = 0 for g 2 EerZ0. If V splits as a sum of 1-dimensional irreducible Z0-representationsf(thisioccurs wh* *enever b is hermitian, or is symplectic or symmetric and 4fi|K*|), then none of these ir* *reducible summands supports a form of the same type as b. Hence in this case, V splits * *as a sum of of eE-representations (U U0), whose characters ØU and ØU0 are such t* *hat ØU(g) = ØU0(g) if g 2 Z and ØU(g) = -ØU0(g) if g 2 Z0r Z. This shows that Ø = ØV 44 BOB OLIVER vanishes on Z0r Z, and hence that (a) holds in this case. Also, since Autf* *fl(Ee), (b) holds by Lemma 7.1(b). Now assume that the Z0-irreducible summands of V are 2-dimensional (and still [Z0:Z] = 2). Since the character of any such irreducible representation vanish* *es on Z0r Z, this finishes the proof of (a). Hence Aut ffl(Ee), and so (b) follo* *ws from Lemma 7.1(a) when G is a linear group (this case does not occur when G is unita* *ry). So assume b is symplectic or symmetric, and hence that |Z0| = 4 (and |K*| 2 (mod* * 4)). In these cases, each irreducible summand of V extends to irreducible representa* *tions of D8xC2X and of Q8xC2X, one of which supports a symplectic form and the other a symmetric (quadratic) form. Since any such form on an irreducible summand is un* *ique up to scalar multiple, this shows that each irreducible summand of V can be ext* *ended to an action of one of the groups D8 xC2 X or Q8 xC2 X which preserves the form* * b. In particular, the involution ff 2 which is the identity on X and sends ff(g)* * = -g for g 2 Z0r Z does lie in AutG (Ee); and this finishes the proof of (b). We are now ready to show that all classical groups of Lie type in odd charact* *eristic lie in L 2(2). Theorem 7.4. Let q be an odd prime power, and let L be one of the simple groups P SLn(q) (n 2), P SUn(q) (n 3), P Sp2n(q) (n 2), or P n(q) (n 5). Then E 2(L ; 2) = ?, and hence L 2 L 2(2). __ __ __ Proof.Write L = [G , G], where G = PAut(V, b), V is a vector space of dimension* * n or 2n over the field K = Fq or Fq2and b is a symplectic, quadratic, hermitian, or * *trivial form. Since P SL4(q) ~=P +6(q), P SU4(q) ~=P -6(q), and P Sp4(q) ~=P 5(q), w* *e can assume that dimK (V ) 6= 4. Set G = Aut(V, b), Z = Z(G) and eL=_[G, G]. Thus G is one of the groups GLn(q* *),_ GUn(q), Sp2n(q), or GOn (q). Also, G = G=Z and L = eL=(Z \ eL). Let ß : G ___! * * G be the projection. Fix E 2 E 2(L ; 2), and set eE= ß-1(E). Define fi = ff 2 AutG (Ee) fiff Id (mod { Id}) . Set fi Ee0= Z. 1(Z(Ee)), eE1= g 2 eE0fiff(g) = g 8ff 2 , and Ei= ß(Eei). By Lemma 7.3, together with the definition of eE1, fi = ff 2 Aut(Ee) fiff|Ee1= Id, ff Id (mod { Id}) ~=Hom (E=E1, { 1}).(1) So if we let Ø: eE___! C be the character of V as an eE-representation, then Ø(g) = 0 for all g 2 eEreE1. (2) Let A Aut (E) be the subgroup of all automorphisms which induce the identity on E1, E0=E1, and E=E0. Then A Aut _G(E) by (1) and Lemma 7.2, and is a normal subgroup since E0 and E1 are both AutG_(E)-invariant subgroups of E. Als* *o, A is a 2-group (see [Gor , Corollary 5.3.3]), so A O2(Aut _G(E)), and A \ L =* * 1 __ since O2(Aut L(E)) =_1 (E is pivotal by Lemma 4.4(c)). Either |G =L| 4 (if G is orthogonal), or G =L is cyclic. Hence A is cyclic or of order 4. Also, rk(* *E) 4 EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 45 by Proposition 4.6(b), and this is easily seen to imply that two of the three g* *roups E1, E0=E1, and E=E0 must vanish. So we are reduced to considering the three cas* *es (E0, E1) = (1, 1), (E, 1), or (E, E). Case 1: Assume first that E0 = 1. Then eE= eP~= Z0xC2 X, where Z0 is cyclic, [Z0:Z] 2, and X is an extraspecial 2-group. Set 2m = rk(X=Z(X)). Since rk(E) * * 4, we must have 2m 4. Let Aut(E) be the group of automorphisms which lift to automorphisms of eE which are the identity on Z. Thus ~=Sp2m(2) if |Z0| 4, and ~=SO2m(2) othe* *rwise. By (2), the character Ø of eEon V vanishes on eErZ, and hence Autffl(Ee). S* *o by Lemma 7.1(c), AutG_(E) has index at most two in . Hence if m 3, then AutL(E)* * is isomorphic to Sp2m(2), SO2m(2), or 2m(2), with the usual action on eE=Z0~= (Z=* *2)2m; and by Proposition 6.4, E cannot lie in E 2(L ; 2). If m = 2, and = Sp4(2) ~=* * 6 or SO-4(2) ~= 5, then again, Aut L(E) has index at most two in , and E =2E 2(L ; * *2) by Proposition 4.7(b). (The orthogonal action of SO-4(2) ~= 5 on (Z=2)4 is the permutation action modulo its fixed component.) It remains to consider the case where ~= SO+4(2) ~= 3 o C2, and thus where eE~= D8 xC2 D8 ~= Q8 xC2 Q8. This group has a unique irreducible representation W on which its central involution acts via (- Id), a representation of dimensio* *n four. Since we are assuming dim(V ) 6= 4, we must have V ~=W kfor some k 2. Then CL(E) = (CeL(Ee)=Z)xE, and hence P 2 Syl2(CL(E)) has the form P = P 0xE for some 2-group P 0. Also, E = 1(Z(P )) by Proposition 4.4(a), so P 0= 1, and CeL(Ee)=* *Z has odd order. But if k > 1, then either V splits nontrivially as an orthogonal dir* *ect sum, in which case P 0contains the automorphisms which are the identity on one summa* *nd and - Idon the others; or k = 2, G = Sp8(q), and CG(Ee) ~=Sp2(q) ~=SL2(q). Thus P 06= 1 in all cases, and hence no subgroup of this type can be in E 2(L ; 2). Case 2: Assume E1 = 1 and E0 = E. Write eE= ZxT where T is elementary abelian. By Lemma 7.2, applied with as in (1), any transvection ff 2 Aut(E) (any invol* *ution which fixes an index two subgroup) lies in AutG_(E). Hence AutG_(E) = Aut(E), s* *ince the transvections generate the full automorphism group (cf. [Di, xx6,10,30]). * * Since Aut(E) ~=GLn(2) is a perfect group for n 3, this shows that AutL (E) = Aut(E). Hence E =2E 2(L ; 2) by Proposition 4.7(a). Case 3: Now assume E1 = E. In this case, CL(E) is the image in L of CeL(Ee), and hence Ee must contain the center of CeL(Ee). Write Ee = Z x T where T is e* *le- L k mentary abelian, and let V = i=1Vi be the decomposition into eigenspaces for * *the characters of T . This is an orthogonal decomposition, and CG(Ee) is the produ* *ct of the groups Aut(Vi, b|Vi). In particular, since E = 1(Z(P )) for some P 2 Syl2(* *CL(E)) (Proposition 4.4(a)), eEcontains the center of CG(Ee), and thus fi eE ' 2 eLfi'|Vi= Id8i . It follows that AutG(Ee) (or its image in Aut(E)) is a product of symmetric gro* *ups: one for each isomorphism class of pairs (Vi, b|Vi). But this is impossible for E 2 * *E 2(L ; 2) by Proposition 4.8. 46 BOB OLIVER 8.Exceptional groups of Lie type in odd characteristic In addition to the five families of Chevalley groups G2(q), F4(q), and En(q),* * it remains to consider the twisted groups 2G2(32k+1), 3D4(q), and 2E6(q) for odd q. The fo* *llowing cases are easy. Proposition 8.1. Assume L is one of the groups G2(q) or 3D4(q) for any odd prime power q, or 2G2(32k+1) for some k 1. Then L 2 L 2(2). _ Proof.By [Gr2 , Theorem 6.1], rk2(G2(F q)) = 3. Hence G2(q) and 2G2(q) have 2-r* *ank at most 3 (in fact, equal to three in all cases). Furthermore, the tables of or* *ders of the groups of Lie type in [Ca1 ] or [GLS3 ] show that for odd q, [3D4(q):G2(q)] = q6(q8 + q4 + 1) is odd, and thus rk2(3D4(q)) = rk2(G2(q)). So E 2(L ; 2) = ? by Proposition 4.6* *(b), and hence L 2 L 2(2) by Proposition 4.2. The groups F4(q) are almost as easy to handle. Proposition 8.2. Assume, for some odd prime power q, that L = F4(q). Then R 2(L ; 2) = ?, and L 2 L 2(2). _ Proof.We regard L as a subgroup of G = F4(F q). There are two conjugacy classes* * of involutions in G, denoted in [Gr2 ] as being of type 2A or 2B (see [Gr2 , Table* * VI]). By [Gr2 , Theorem 7.3], G contains a unique conjugacy class of maximal elementa* *ry abelian 2-subgroups, represented by E5 = T(2)x <`>, where T is a maximal torus,* * T(2) is its 2-torsion subgroup, and ` 2 NG(T ) is an element which inverts T . Furth* *ermore, the elements of type 2B in E5 form (together with the identity) a subgroup E2 * * E5 of rank 2. Thus for any elementary abelian 2-subgroup E L, there is a subgroup E0 E such that all involutions in E0 are of type 2B in G and all involutions in Er E* *0 are of type 2A, and such that rk(E0) 2 and rk(E=E0) 3. In particular, E0is NAut(G)* *(E)- invariant, and thus NAut(L)(E)-invariant. Hence E 2(L ; 2) = ? by Proposition 4* *.6(b), and so L 2 L 2(2) by Proposition 4.2. In order to deal with_the remaining cases, we need to look more closely at the algebraic groups over Fq of which they are subgroups. Our general references fo* *r the properties of algebraic groups are [Hum ] and [Ca2 ]. Note in particular that * *connected algebraic groups always have maximal tori (products of copies of F *) which are* * unique up to conjugacy [Hum , x21.3]. For an arbitrary algebraic group G, we let G0 d* *enote the connected component of the identity: a normal subgroup of finite index [Hum , * *x7.3]. A connected algebraic group over an algebraically closed field F is reductive* * if it has no nontrivial normal unipotent subgroup; this is equivalent to its being th* *e central product over a finite group of a semisimple group and a torus [Hum , xx19.5 & * *27.5]. We first note the following very elementary and well known results about centra* *lizers and normalizers of subgroups of a maximal torus in a reductive group. Lemma 8.3. Let G be an algebraic group over an algebraically closed field F who* *se identity component G0 is reductive. Fix a maximal torus T G, and set W = NG(T )=T (regarded as a group acting on T ). Let be the set of roots of G, re* *garded EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 47 as elements of Hom (T, F *). For each ff 2 , let Xff G denote the correspondi* *ng root subgroup (Xff~=F ). Then for any subgroup H T , CG(H)0 = is a reductive group, with root system {ff 2 | ff(s) = 1, all s 2 H}. Also, CG(H) = CG(H)0.{wT 2 W | w(s) = s all s 2 H}; AutG (H) = AutW (H); and two elements x, y 2 T are CG(H)-conjugate if and only * *if they are CW (H)-conjugate. Proof.The description of CG(H)0 in (a) is shown in [Ca2 , Theorem 3.5.3] when G* * is connected and H = is cyclic, and the proof given there also applies in the * *more general case. Since T is a maximal torus in CG(H)0, and all maximal tori are conjugate, each element of NG(H) conjugates T to another maximal torus, and hence CG(H) = CG(H)0.CNG(T)(H) and NG(H) = CG(H)0.NNG(T)(H). This proves the descriptions of CG(H) and AutG (H). Similarly, for any x, y 2 T and g 2 CG(H) such that y = gxg* *-1, T and gT g-1 are two maximal tori of the reductive group CG(H, y)0, hence conju* *gate in CG(H, y)0, so there is a 2 CG(H, y) such that a(gT g-1)a-1 = T , and ag 2 CNG(T* *)(H) conjugates x into y. We adopt the terminology used in [GLS3 ], and write Steinberg endomorphism to mean a surjective algebraic endomorphism of an algebraic group whose fixed subg* *roup is finite. All finite simple groups of Lie type can be constructed as (commuta* *tor subgroups of) fixed subgroups of Steinberg endomorphisms. The following result* * is one of the key properties of these endomorphisms. Proposition 8.4._(Lang-Steinberg theorem) Fix a prime p and a connected algebra* *ic group G over Fp. Let oe be any Steinberg endomorphism of G. Then every element * *of G is of the form x-1oe(x) for some x 2 G. Proof.See [St, Theorem 10.1]. The proof is also sketched in [Ca2 , x1.17]. The following proposition is a special case of [GLS3 , Theorem 2.1.5]. It des* *cribes, in many cases, the relationship between conjugacy classes and normalizers in a con* *nected algebraic group with those in the subgroup fixed by a Steinberg endomorphism. _ Proposition 8.5. Let G be any connected algebraic group over Fq. Fix a Steinbe* *rg endomorphism oe of G. Let H CG(oe) be any subgroup, and let H be the set of __ * * def CG(oe)-conjugacy classes of subgroups G-conjugate to H. Let_NG(H) act on_C G(H)* * = ß0(CG(H)) by sending g to xgoe(x)-1 (for x 2 NG(H)). Let C G(H)=N and C G(H)=C denote sets of orbits of the NG(H)- and CG(H)-actions, respectively. Then there* * is a bijection ~= __ ! : H -------! C G(H)=N, where !(xHx-1) = [x-1oe(x)] for x 2 G such that xHx-1 CG(oe). Also, for any H0 2_H,_Aut CG(ff)(H0) is the stabilizer of !(H0) under the action of Aut G(H) * *on the set C G(H))=C of CG(H)-orbits. 48 BOB OLIVER Proof.Note first that NG(H) does act on CG(H) and hence on ß0(CG(H)). If g 2 NG(H) and x 2 CG(H), then for all h 2 H, -1 -1 -1 -1 -1 -1 -1 -1 gxoe(g) .h. gxoe(g) = gxoe(g hg)x g = g(g hg)g = h (since g-1hg 2 H CG(oe)), and hence gxoe(g)-1 2 CG(H). If x 2 G is such that xHx-1 CG(oe), then cff(x)is equal to cx on H, and thus x-1oe(x) 2 CG(H). We first check that ! is well defined; i.e., that the NG(H)-o* *rbit of x-1oe(x) depends only on the CG(oe)-conjugacy class of xHx-1. Assume first that xHx-1 = yHy-1 CG(oe), and set g = y-1x 2 NG(H). Thus x = yg, x-1oe(x) = g-1 y-1oe(y) oe(g), __ and so [x-1oe(x)] = [y-1oe(y)] 2 C G(H)=N. Finally, if x 2 G is such that xHx-1 CG(oe), and g 2 CG(oe), then (gx)-1oe(gx) = x-1(g-1oe(g))oe(x) = x-1oe(x), and thus !(xHx-1) = !(gxHx-1g-1). This shows that ! is well defined. To see that ! is onto, fix some z 2 CG(H).* * By the Lang-Steinberg theorem, there is x 2 G such that x-1oe(x) = z. Then conjuga* *tion by x and by oe(x) are equal on H (since x-1oe(x) centralizes H), so xHx-1 CG(* *oe), and [z] = !(xHx-1) 2 Im(!). To see that ! is injective, it now suffices to consider the case where x, y 2* * G are such that x-1oe(x) y-1oe(y) (mod CG(H)0). Let z 2 CG(H)0 be such that x-1oe(x* *) = y-1oe(y).z. Then x-1oe(x) = y-1 oe(y)zoe(y)-1 oe(y) so that (xy-1)-1oe(xy-1) = oe(y)zoe(y)-1 2 CG(yHy-1)0. By the Lang-Steinberg theorem, (xy-1)-1oe(xy-1) = g-1oe(g) for some element g 2 CG(yHy-1)0. Then xy-1g-1 2 CG(oe), and so xHx-1 and yHy-1 are CG(oe)-conjugate. It remains to describe Aut CG(ff)(H0) for any H0 = xHx-1 2 H. Consider the monomorphism fl : Aut CG(ff)(xHx-1) -------! Aut G(H) ~=NG(H)=CG(H) (1) which_sends cxgx-1to cg. We must show that Im (fl) is the stabilizer of [x-1oe* *(x)] 2 CG (H)=C under the action of AutG (H). Assume first that cg 2 Aut G(H) stabilizes [x-1oe(x)]. Upon replacing g by a* *g for some_appropriate a 2 CG(H), we can assume that g fixes the class of x-1oe(x) in CG (H) = ß0(CG(H)) itself. Thus -1 -1 -1 -1 -1 -1 0 g x oe(x) oe(g) = (xg ) oe(xg ) x oe(x) (mod CG(H) ), and hence (as already shown above) there is z 2 CG(xHx-1) such that x(xg-1)-1z-* *1 2 CG(oe). Thus, xgx-1z-1 2 NCG(ff)(xHx-1), and fl sends conjugation by this eleme* *nt to cg. Conversely, for any g 2 NCG(ff)(xHx-1), -1 -1 -1 -1 -1 -1 (x-1gx). x oe(x) .oe(x gx) = x goe(g) oe(x) = x oe(x); and hence x-1gx stabilizes [x-1oe(x)]. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 49 For example, if H = T(2)is the 2-torsion in a oe-invariant maximal torus T * *G (and H CG(oe)), then CG(H) is generated by T , together with the element of W = N(* *T )=T which inverts T if there is such an element. Hence ß0(CG(H)) has at most two el* *ements. So either there are two CG(oe)-conjugacy classes of subgroups G-conjugate to H * *(if some element of W inverts T ), or there is just one such class (if no element of W i* *nverts T ). Proposition 8.6. Let p be any prime, and let G be a connected reductive algebra* *ic group over an algebraically closed field F of characteristic 6= p. Then for an* *y finite p-subgroup P G, ß0(CG(P )) is a p-group, and the identity connected component CG(P )0 of the centralizer is also a reductive algebraic group over F. Proof.Let P 0C P be a subgroup of index p; we can assume inductively that the l* *emma holds for P 0. Thus CG(P 0)0 is reductive and has p-power index in CG(P 0). F* *ix any g 2 P rP 0. Then g acts on CG(P 0) as an algebraic automorphism, and we must sh* *ow that the fixed point set of its action on CG(P 0)0 has reductive identity compo* *nent, and has p-power index in the full fixed point set. In other words, it suffices to c* *onsider the case where P = is cyclic, but where instead of ff being an element of G we* * only assume that it is an algebraic automorphism of G of p-power order. Since p is p* *rime to char(F), ff is a semisimple automorphism. Let T C G be the maximal normal toral subgroup, and let H C G be the maximal normal semisimple subgroup. Let eH be the universal cover of H. Set K = Ker[T x eH--_! G], where _ is the sum of the inclusion of the torus and the projection * *of eH onto H. By [St, 9.16], ff lifts to a unique automorphism of eH, which can be as* *sumed also to have p-power order (otherwise, just replace it by an appropriate power)* *. We thus get an exact sequence 1 --! CK (ff) ---! CT(ff) x CHe(ff) ---! CG(ff) ---! H1(; K). (1) The last group is a finite p-group, since K is finite and is a p-group. Al* *so, CHe(ff) is connected and reductive by [St, Theorem 8.1] (and this is the deep result wh* *ich lies behind the proposition). The identity component of CT(ff) is, of course, a toru* *s; and is the image of the norm map N(since this image is connected and has finite* * index in the centralizer). Thus ß0(CT(ff)) = (T ff)=(N.T ) ~=bH0(; T ) is a p-group. Together with the exact sequence (1), this finishes the proof th* *at ß0(CG(ff)) is a p-group, and that CG(ff)0 is reductive. The next proposition will be used to show that certain elementary abelian sub* *groups are not pivotal. Note that the condition on oe2 in the statement holds in all s* *ituations which occur for finite groups of Lie type, except those involving the triality * *automor- phism and 3D4(q). In general, when T is a maximal torus in an algebraic group G* *, we let T(2)denote the subgroup of elements of order 2. _ Proposition 8.7. Let G be a connected reductive algebraic group over Fq, where q is odd, and fix a maximal torus T of G. Let oe be a Steinberg endomorphism of G such that oe(T ) = T , and such that oe2 is the identity on W = NG(T )=T and on* * T(2). Let E CT(oe) be an elementary abelian 2-subgroup. Assume there is an involut* *ion x 2 T rE such that the orbit of x under the CW (E)-action on T has odd order. T* *hen no subgroup_of CG(oe) which is G-conjugate to E is pivotal in CG(oe). More gene* *rally, if E E is also elementary abelian, and is such that x is not CG(E)-conjugate * *to any 50 BOB OLIVER __ element of E , then for any L C CG(oe)_which_contains {gxg-1 | g 2 G} \ CG(oe),* * no subgroup of L which is G-conjugate to E is pivotal in L. Proof.We first show, for any E0 L C CG(oe) which is G-conjugate to E, that th* *ere are g 2 G and k 1 which satisfy the following conditions: k (a)E0= gEg-1 and g-1oe2 (g) 2 T ; k 0def -1 0def 0 * * 0 (b)oe2 leaves T = gT g invariant and acts via the identity on W = NG(T )=T* * and on T 0(2); (c)Aut CG(ff2k)(T 0) = AutG (T 0); and (d)there is a CL(E0)-conjugacy class X0 of odd order whose elements are all G- conjugate to x0def=gxg-1. __ __ Assume these have been shown,_and_let E 0 L be a subgroup G-conjugate to E . Choose h 2 G such that E 0= hE h-1, and set E0 = hEh-1. By (c), we can replace g (the element satisfying (a) and (b)) by ag for an appropriate element a 2 NCG(f* *f2k)(T 0), __ __ __ to arrange that h 2 CG(E0).g. Then E 0and gE g-1 are CG(E0)-conjugate;_and E 0 contains no elements CG(E0)-conjugate_to x0 = gxg-1 since_E contains no elemen* *ts CG(E)-conjugate to x. Hence E 0\ X0 = ? by (d), and E 0cannot be pivotal in L by Proposition 4.4(f). It remains to show points (a-d). Consider the homomorphism j CW (E) --- --i ß0(CG(E)), which is surjective by Lemma 8.3. Since oe acts on W with order 2, and since CW* * (E) has an odd number of Sylow 2-subgroups, we can choose S0 2 Syl2(CW (E)) which is oe-invariant. Since ß0(CG(E)) is a 2-group (Proposition 8.6), æ|S0 is also s* *urjective. Using Proposition 8.5, we can now choose g such that E0 = gEg-1, and such that g-1oe(g) 2 aT for some aT 2 S0. Then for k sufficiently large, k 2 2k-1 2k-1 g-1oe2 (g) 2 a.oe(a).oe (a). .o.e (a) T = a.oe(a) T = T. This proves (a), and (b) then follows easily. Furthermore, upon replacing k by * *k + 1, k 0 we can assume that oe2 acts via the identity on the subgroup H T generated * *by elements of order 4 in T 0. k 0 0 0def -1 Set Gk = CG(oe2 ) for short. Let X be the CGk(E )-orbit of x = gxg . By Lem* *ma 8.3, any two elements of X0\ T 0are CW0(E0)-conjugate. Also, by the Lang-Steinb* *erg theorem, each coset of T 0in NG(T 0) has representatives in Gk, and thus X0\ T * *0is the full CW0(E0)-orbit of x0. Since the CW (E)-orbit of x has odd order by assumpti* *on, this shows that X0\ T 0has odd order. Since X0\ T 0= X0\ CG(H) is the fixed set of an action of the 2-group H on X0, X0must also have odd order. Note also that Aut G(E0) = AutW0(E0) by Lemma 8.3, and hence Aut G(E0) = AutGk(E0) since each coset of T 0in NG(T 0) has representatives in Gk. This pr* *oves (c). Since |X0| is odd, and since oe acts on X0 CGk(E0) with order 2k, the fixed * *point set X0\ CG(oe) of this action also has odd order. Also, by assumption, X0\ CG(o* *e) EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 51 L C CG(oe), and is a union of CL(E0)-conjugacy classes. So there is a CL(E0)-co* *njugacy class X0 X0\ L of odd order, and this proves (d). We are now ready to consider the remaining exceptional groups En(q), and 2E6(* *q). We adopt some of the notation used by Griess in [Gr2 ]. In particular, 2A and 2* *B will be used to denote conjugacy classes of elements of order 2. _ _ _ In all cases, we let E6(F q),_E7(F q), or E8(F q) denote the adjoint (centerf* *ree)_forms of these groups,_and let eEn(F q) denote their universal_covers. Thus eE6(F q) * *is a 3-fold_ cover of E6(F q) (when q is not a power of 3),_and eE7(F q) is a 2-fold cover o* *f E7(F q). In all cases, by [Gr2 , Lemma 2.16], if T eEn(F q) is a maximal torus, there is * *a quadratic form q: T(2)___! F2 which sends one of the conjugacy classes to 1 and the othe* *r to 0, and the Weyl group acts on T(2)as the full orthogonal group for the form q. Proposition 8.8. Assume, for some odd prime power q, that L is one of the simple groups E6(q) or 2E6(q). Then R 2(E6(q) ; 2) = ?, and hence L 2 L 2(2). _ Proof.Set G = E6(F q) (in adjoint form, with trivial center). Let _q 2 Aut (G)* * be the field automorphism induced by (x_7! xq), and let ø 2 Aut (G) be the graph a* *u- tomorphism with fixed subgroup F4(F q). Set oe = _q if L ~= E6(q) or oe = _q O* *ø if L ~= 2E6(q). In either case, we let L be the commutator subgroup of CG(oe). N* *ote that CG(oe) = Inndiag(L) contains L with index (3, q - 1) if L = E6(q), or with* * index (3, q + 1) if L = 2E6(q). Fix a maximal torus T G upon which oe acts via (x 7! xq) if L = E6(q), or v* *ia (x 7! x-q) if L = 2E6(q). In the latter case, this can be found using the Lang-* *Steinberg theorem: if T 0is the "standard" torus upon which _q acts via (x 7! xq) and ø a* *cts with fixed subtorus of rank 4, then there is wT 02 N(T 0)=T 0such that cw Oø(x)* * = x-1 for all x 2 T 0, w = g-1oe(g) for some g 2 G, and we can take T = gT 0g-1. Thus in either case, T(2) L. By [Gr2 , Lemma 2.16], there is a nonsingular q* *uadratic form q: T(2)--! F2 such that q(x) = 1 for x of type 2A and q(x) = 0 for x of ty* *pe 2B, and such that the Weyl group action on T(2)is that of the orthogonal group O(T(2), q) ~=SO-6(2). Since the fundamental group of G has order prime to 2, [C* *a2 , Proposition 3.7.3] applies to show that L has the same two classes of involutio* *ns. By [Gr2_, Theorem 8.2], there is a unique maximal elementary abelian 2-subgro* *up W5 E6(F q)_which is not_contained in a maximal torus, and W5 is contained_in a subgroup F4(F q) E6(F q). By [Gr2 , Lemma 2.16(i)], involutions in F4(F q) o* *f types 2A and_2B are sent under this embedding to involutions of types 2A and 2B in E6(F q). So by [Gr2 , Theorem 7.3(ii)], the elements of type 2B in W5 (together* * with the identity) form a subgroup W2 W5 of rank 2. Thus for any elementary abelia* *n 2- subgroup E L which is not contained in a maximal torus of G, there is an NAut* *(L)(E)- invariant subgroup E0 E such that rk(E0) 2 and rk(E=E0) 3, and E =2E 2(L ;* * 2) by Proposition 4.6(b). Now fix some E 2 E 2(L ; 2). Since E cannot be G-conjugate to a subgroup of W* *5, it must be G-conjugate to a subgroup E0 T(2). Also, E is pivotal (Proposition 4.4* *(c)), and rk(E) 4 by Proposition 4.6(b). By Lemma 8.3(b), Aut G(E) ~= AutG(E0) ~= SO(E, q), since every element of AutG (E0) is the restriction of the action of * *an element of the Weyl group. 52 BOB OLIVER Assume rk(E) = 4. If q|E is singular, then E \ E? is a proper subgroup of E w* *hich is NAut(L)(E)-invariant, so E =2E 2(L ; 2) by Proposition 4.6(b). So we assume* * q is nonsingular on E, and hence on E0 T(2). Then T(2)= E0x E0?, E0? is CW (E0)- invariant, and hence there is 1 6= x 2 E0? whose CW (E0)-orbit has odd order. * *By Proposition 8.7, no subgroup of L which is G-conjugate to E0 is pivotal in L. * *In particular, E =2E 2(L ; 2) in this case. Finally, if rk(E) 5, then we are in one of the following situations: either (i)rk(E) = 6 and AutG (E) ~=SO-6(2); or (ii)rk(E) = 5, q|E\E? = 0, and AutG (E) ~=24:SO-4(2); or (iii)rk(E) = 5, q|E\E? 6= 0, and AutG (E) ~= 5(2) ~=Sp4(2) ~= 6. If any element of W fixes a rank five subgroup E0 T(2), then it must be an ort* *hogonal transvection, and hence E0 must be of type (iii) above_(see [Di, x19]). Thus th* *e kernel of any root of E6 (regarded as an element of Hom (T, F*q)), when restricted to * *T(2), must also be a subgroup of type (iii). So by Lemma 8.3, CG(E) = T if E has type (i)* * or (ii), and CG(E) is connected if E has type (iii). In all cases, AutCG(ff)(E) = * *AutG (E) by Proposition 8.5, and hence Aut L(E) = Aut G(E) since Aut G(E) has no normal subgroup of index 3. So E =2E 2(L ; 2): by Proposition 4.6(c) in the first and * *third cases (the Sylow 2-subgroups of AutL(E) are neither dihedral nor semidihedral),* * or by Proposition 4.4(c) in the second case (E is not pivotal). _ Let_eE7(q) _eE7(F q) denote the universal groups, with center Z of order 2, * *and set E7(F q) = eE7(F q)=Z and E7(q) = eE7(q)=Z. Proposition 8.9. Assume, for some odd prime power q, that L = E7(q). Then R 2(L ; 2) = ?, and hence L 2 L 2(2). _ Proof.Set eG= eE7(F q), let z 2 Z(Ge) be the central involution, and set G = eG* *= = _ def E7(F q). Fix a Steinberg endomorphism oe of eG such that CGe(oe) = eL = Ee7(q)* *. We also let oe denote the induced endomorphism of G. Note, however, that CG(oe) = Inndiag(L) ~=L:2 is the extension of L ~=E7(q) by its diagonal automorphisms. L* *et eT eGbe a oe-invariant maximal torus, and set T = eT=. _ By [Gr2 , Table VI], the group eE7(F q) has two conjugacy classes of noncentr* *al invo- lutions, referred to as types 2B and 2C, which are exchanged under multiplicati* *on by z. Define q: eT(2)--!F2 by setting q(x) = 1 if x = z or x is of type 2B, and q(* *x) = 0 if x = 0 or x is of type 2C. Then q is a quadratic form [Gr2 , Lemma 2.16], and W (E7) = SO(T(2), q) x C2 ~=SO7(2) x C2. In particular, this means that no automorphism of eLcan switch the two classes * *2B and 2C _ since q would then no longer be a quadratic form. The symplectic form associated to q induces a symplectic form b on the subgro* *up T(02) T(2)of elements which lift to involutions. To describe this, for each 1 * *6= x 2 T(02), let ex2 eT(2)be the (unique) lifting of x of type 2C (and set e1= 1). Then b(x,* * y) = 0 if ex.ey= fxy, and b(x, y) = 1 if ex.ey= z.fxy. It will be convenient to extend th* *is definition to any elementary abelian subgroup E L all of whose elements lift to involuti* *ons; although of course b will no longer be bilinear in general if E is not toral. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 53 We note that no element of CG(oe)r L lifts to an involution in eG. Since if t* *here were such an element, lifting to x 2 eG, then x and xz would be exchanged by oe, and* * hence oe would send all involutions of type 2B to involutions of type 2C and vice ver* *sa. That would imply that the only involution in eL~= eE7(q) is the central element, whi* *ch is clearly not true. By [Gr2 , Theorem 9.8], G contains two conjugacy classes of maximal elementary abelian 2-subgroups, both nontoral: M8 of rank 8 and M7 of rank 7. These lift* * to centric subgroups fM8 ~=Q8 x 26 and fM7 ~=Q8 x 25 in eG. Also, M8 = T(2).<`>, * *the extension of 2-torsion in a_maximal torus by an involution in NG(T ) which inve* *rts the torus; while fM7 Q8 x F4(F q) eG. Fix E 2 E 2(L ; 2), and P 2 Sylp(CL(E)). Thus P 2 R 2(L ; 2) and E = 1(Z(P )* *). Let E0 E be the subgroup of elements of E which lift to involutions in eL(a su* *bgroup by the above remarks, and clearly NAut(L)(E)-invariant). We now consider the di* *fferent possibilities. Case 1: Assume E is G-conjugate to a subgroup_of M7. As noted_above, fM7= QxV , where Q ~=Q8, and V ~= 25 is a subgroup of F4(F q) eE7(F q)._ There is a subg* *roup V2 V such that the involutions in V2 lie in one class in F4(F q) and those in* * V rV2 in the other ([Gr2_, Theorem 7.3]). Also, any index two subgroup of V containin* *g V2 is toral in F4(F q), hence in G, so the function q as defined above is quadrati* *c on this subgroup. Since eT(2)contains no rank four isotropic subspace, this is possibl* *e only if the involutions in V2 have type 2C in G and those in V rV2 have type 2B. Thus b: V x V ___! F2 takes the form b(x, y) = 1 if x, y generate a rank two subgro* *up of V=V2 and b(x, y) = 0 otherwise. Now set E00= {x 2 E0| b(x, E0) = 0}; this is again a NAut(L)(E)-invariant sub* *group of E. Upon identifying E0 with a subgroup of V , we see that either E00= E0 and rk(E0) 3, or rk(E00) 2 and rk(E0=E00) 3. In both cases, E 2= E 2(L ; 2) * *by Proposition 4.6(b). Case 2: Assume E is G-conjugate to a subgroup of M8 = T(2).<`> as above. In particular, E0 is a toral subgroup, and hence b is a symplectic form on E0 whic* *h is invariant under the action of NAut(L)(E). By Proposition 4.6(b) again, rk(E0) * * 4; and if rk(E0) = 4 then b must be nonsingular on E0. By [Gr2 , Theorem 9.8], Aut G(M8) ~= 27:Sp6(2). The normal subgroup 27 is the group of all automorphisms which are the identity on T(2)(induced by conjugatio* *n by elements of order 4 in T ). Thus AutG (E) ~=Aut (E0, b) if E is toral (use Lemm* *a 8.3). If E is not toral, then Aut G(E) surjects onto Aut (E0, b) with kernel an eleme* *ntary abelian 2-group. Case 2a: If rk(E0) = 6, then AutG (E) surjects onto AutG (E0, b) ~=Sp6(2). Als* *o, E0 is conjugate to eT(2)=, and CG(Te(2)=) = T .<`> by Lemma 8.3. Hence either o E is toral, CG(E) is a maximal torus extended by the Weyl element which inve* *rts it, and AutG (E) ~=Sp6(2); or o E is not toral, CG(E) ~=28, and AutG (E) ~=2k:Sp6(2) where k = rk(E) - 1. In the first case, AutL(E) ~=Sp6(2) by Proposition 8.5. 54 BOB OLIVER Assume E is not toral. We look at the action of AutG (E) on CG(E), where x ac* *ts by g 7! xgoe(x)-1. We assume that T is the "standard" torus, on which oe acts by x* * 7! xq. Thus either oe fixes all x 2 T of order 4, or oe(x) = x-1 for all such x. Also,* * all Weyl group elements (cosets of T in NG(T )) contain elements in CG(oe). We must determine the isotropy subgroups of this action, and it suffices to d* *o this when E M8 = T(2).<`>. We have CG(E) = M8, and O2(Aut G(E)) is the group of automorphisms which are induced by conjugation by elements of order 4 in T . Thus O2(Aut G(E)) acts freely on one of the cosets T(2)or T(2).` (with one or t* *wo orbits), and on the other coset with the fixed action. Hence for each isotropy * *subgroup H AutG (E) of this action, either O2(H) 6= 1, or (since Sp6(2) has no subgrou* *p of index 2) H ~=Sp6(2). Thus by Proposition 8.5 again, either O2(Aut L(E)) 6= 1, and E is not pivotal* * (contra- dicting Proposition 4.4(c)); or AutL (E) ~=Sp6(2), and E =2E 2(L ; 2) by Propos* *ition 6.4. Case 2b: If rk(E0) = 5, then E00def=E0\E0? has rank one and is NAut(L)(E)-inva* *riant, and NL(E) (and NL(P )) are contained in CL(E00) ~=SL2(q) xC2 SSpin+12(q) (see [Gr2 , Table VI] for the centralizer of an involution of type 2B or 2C). A* *lso, NAut(L)(E) CAut(L)(E00). Hence by Proposition 4.9(b), applied with H = CL(E0* *0), H0 = SSpin+12(q), and K = SL2(q), P K=K 2 R2(P +12(q) ; 2). But this is imposs* *ible, since R2(P +12(q) ; 2) = ? by Theorem 7.4. Case 2c: Assume rk(E0) = 4. Let V T(2)be the subgroup of elements which lift to involutions in eG, and let V4 V be a rank 4 subgroup on which b is nonsing* *ular; thus G-conjugate to E0. Fix an involution x 2 V4? V . The orbit of x, under* * the action of those Weyl group elements which fix V4, has order 3; and thus the gro* *up of Weyl group elements which fix has index 3 in the group of those which f* *ix V4. By construction, x is not G-conjugate to any element of Er E0. Also, as noted a* *bove, x is not conjugate to any element of CG(oe)r L since it lifts to an involution * *in eG. So by Proposition 8.7, E is not a pivotal 2-subgroup in L. It remains only to consider the groups E8(q). Proposition 8.10. Assume, for some odd prime power q, that L = E8(q). Then R 2(L ; 2) = ?, and hence L 2 L 2(2). _ Proof.Set G = E8(F q). Fix a Steinberg endomorphism oe of G such that CG(oe) = * *L. Let T G be a oe-invariant maximal torus. _ By [Gr2 , Table VI], the group E8(F q) has two conjugacy classes of involutio* *ns, de- noted types 2A and 2B. Define q: T(2)--! F2 by setting q(x) = 1 if x has type 2A and q(x) = 0 otherwise. Then by [Gr2 , Lemma 2.16], q is a quadratic form. Also* *, if ` 2 NG(T ) is an involution which inverts T , then W=<`T >= AutG (T )={ Id} ~=O(T(2), q) ~=SO+8(2). By [Gr2 , Theorem 2.17], G contains two conjugacy classes of maximal elementa* *ry abelian 2-subgroups, represented by M9 = T(2)x<`> of rank 9 (where ` again inve* *rts T ), and M8 of rank 8. All elements in M9r T(2)are of type 2B. Also, there are subgr* *oups EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 55 V, W M8 of rank five, with intersection V \ W = X of rank two, such that the elements of type 2A in M8 are precisely those in (V [ W )r X. Assume E 2 E 2(L ; 2). Fix P 2 Syl2(CL(E)), so that P 2 R 2(L ; 2) and E = 1(Z(P )) by Proposition 4.4(a). Also, E is pivotal (Proposition 4.4(c)), and * *thus O2(Aut L(E)) = 1. By Proposition 4.4(e), E must contain at least one involutio* *n of type 2B, since these are in the center of any Sylow 2-subgroup of L (there is a* *n odd number of them in any maximal torus since there are exactly 135 nontrivial isot* *ropic elements of T(2)). Also, there must be at least one NAut(L)(E)-irreducible subq* *uotient of E which has rank 4 (Proposition 4.6(b)). We will show that this is impossi* *ble, and thus that E 2(L ; 2) = ?. Case 1: Assume first that there is an NAut(L)(E)-submodule E0 E of rank one. Then NAut(L)(E) CAut(L)(E0), and the centralizer is described as follows in [* *Gr2 , Table VI]: __________________________________|||||| || type| CG(E0) | CL(E0) || ||___|_____________|____________||__|||_||| || 2B| SSpin16(F q)| SSpin+16(q) || ||___|_____________|_____________|||||__||| || 2A| F*qxC2 eE7(F|q)F*qxC2 eE7(q) || ||___|_____________|______________ || We apply Proposition 4.9(b), with H = CL(E0), H0 = SSpin+16(q) or eE7(q), and K* * = CH (H0) ~= C2 or F*q. Then P K=K 2 R 2(H=K ; 2). But we have already seen that R 2(P +16(q) ; 2) and R 2(E7(q) ; 2) are both empty (Theorem 7.4 and Propositi* *on 8.9), and so this is impossible. Case 2: Assume that E is not G-conjugate to a subgroup of M9. Hence E is conta* *ined in M8 up to G-conjugacy; and we assume (replacing M8 if necessary) that E M8. Also, this means E contains involutions of both types. Let V, W M8, with intersection X = V \W of rank two, be as above. If E \X 6* *= 0, then (since rk(E) 4) it is characterized as the largest type 2B pure subgroup* * of E each of whose cosets is all type 2B or all type 2A. Hence E \X is NAut(L)(E)-in* *variant. Thus since E has no rank one invariant submodules (case 1), either E \ X = 0, or E X. Also, since E must have some irreducible component of rank 4, we see t* *hat either E=(E \ X) has rank 6, or it has rank 4 and intersects each of V and W wi* *th rank 2. In the latter case, the map q: E --! F2 which sends involutions of type* * 2A to 1 and other elements to zero is a quadratic form, hence E is toral by [Gr2 ,* * Theorem 9.2], in contradiction to our assumption. If E X and rk(E=X) = 6, then E = M8, and by [Gr2 , Theorem 2.17], CG(E) = E and O2(Aut G(E)) ~=212. Hence O2(H) 6= 1 for each isotropy subgroup H of the ac* *tion of AutG (E) on CG(E), so O2(Aut L(E)) 6= 1 by Proposition 8.5, and E is not piv* *otal in this case. If E \ X = 0 and rk(E) = 6, then E = E1 x E2 where the elements of type 2A are precisely those in (E1[E2)r 1. Let E0 E be a subgroup of rank 4 whose interse* *ctions with E1 and E2 each have rank two. Then q|E0 is quadratic, hence E0 is toral by* * [Gr2 , Theorem 9.2], and is a maximal toral subgroup of E. Choose E00 T(2)which is G-conjugate to E0, and let E0 E00be such that the pair (E0, E00) is G-conjuga* *te to (E, E0). Then CW (E00) leaves E00?invariant, and we can choose 1 6= x 2 E00?who* *se CW (E00)-orbit has odd order. Since E00is a maximal toral subgroup of E0, x is* * not 56 BOB OLIVER CG(E00)-conjugate to any element of E0. Hence by Proposition 8.7, no subgroup o* *f L which is G-conjugate to E0 can be pivotal, and in particular E =2E 2(L ; 2). Case 3: Next assume E is G-conjugate to a subgroup of M9 = T(2)x<`> and contai* *ns involutions of both types. Let E0 E be the subgroup generated by all involuti* *ons of type 2A; this is clearly NAut(L)(E)-invariant, and is a toral subgroup since al* *l elements of M9r T(2)are of type 2B. By inspection, one sees that for any V T(2)which contains elements of both * *types, either V is generated by its elements of type 2A; or rk(V ) 5 and the element* *s of type 2A generate a subgroup V0 V of index 2 within which the involutions of type 2B (with the identity) form a subgroup of index 2. Thus, either E = E0, or [E:E0] * *= 2 and E is not toral, or rk(E0) 4 and there is an NAut(L)(E)-invariant subgroup E00* * E0 of index 2. In particular, no group of this last type contains an irreducible c* *omponent of rank 4. We can thus assume [E:E0] 2, and that either E = E0 or E is not toral. Cons* *ider the quadratic form q (as defined above) on E0, and set E1 = E0? \ E0. If rk(E1)* * = 4, then E0 = E1, q is linear on E0, and (since we are assuming E contains elements* * of type 2A) E2 def=Ker(q|E0) is an NAut(L)(E)-invariant subgroup of E of rank thre* *e. But this would contradict Proposition 4.6(b). Thus rk(E1) < 4, and so (by Proposition 4.6(b) again) we can assume rk(E0=E1)* * 4. Hence either the bilinear form on E0 is nonsingular (E1 = 0); or (since E has no invariant submodule of rank one) rk(E1) = 2, rk(E0=E1) = 4, and all involutions* * in E1 are of type 2B. Case 3a: Assume the bilinear form associated to q is nonsingular on E0 (E1 = * *0), and rk(E) < 8. Let E00 T(2)be G-conjugate to E0, and let E0 E00be such that the pair (E0, E00) is G-conjugate to (E, E0). Then CW (E00) leaves E00?invarian* *t, and we can choose 1 6= x 2 E00?whose CW (E00)-orbit has odd order. If E E0, then * *E0 is not toral and is, so x is not CG(E00)-conjugate to any element of E0. * *Hence by Proposition 8.7, no subgroup of L which is G-conjugate to E0 can be pivotal; an* *d in particular, E =2E 2(L ; 2). Case 3b: Assume rk(E0) = 8; i.e., E is G-conjugate to T(2)or M9. We apply Proposition 8.5 to determine the possibilities for AutL (E). Since ß0(CG(T(2)))* * ~=C2, T(2)restricts to two conjugacy classes in L, both of which have AutL(E0) ~=SO+8* *(2). Now, CG(M9) = M9 and AutG (M9) ~=28:SO+8(2), and O2(Aut G(M9)) acts freely on one of the cosets T(2)or T(2).` and trivially on the other. Thus if E is G-conj* *ugate to M9, then either AutL(E) ~=SO+8(2), or O2(Aut L(E)) 6= 1 (has 2-rank at least 8). Thus, since O2(Aut L(E)) = 1 (E is assumed pivotal), we have AutL(E) ~=SO+8(2* *), and E =2E 2(L ; 2) by Proposition 6.4. Case 3c: Assume rk(E1) = 2, rk(E0=E1) = 4, and all involutions in E1 are of ty* *pe 2B. Each automorphism of E0 which is the identity on E1 and on E0=E1 preserves * *q, and hence lies in AutG (E0). Thus |O2(Aut G(E0))| 28. Also, since E0 contains* * a 2B- pure subgroup of rank 4, CG(E0)0 is a maximal torus (see Case 4), and ß0(CG(E0)* *) ~=22 (generated by the unique involution in SO+8(2) which fixes a rank 6 subgroup of* * this type and the Weyl group element which inverts the torus). So by Proposition 8.* *5, O2(Aut L(E0)) 6= 1, and E0 is not pivotal. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 57 Thus E E0. Set T 0= CG(E0)0: a maximal torus of G. Since ß0(CG(E0)) ~=22 and the elements of Er E0 invert T 0, we have |CG(E)| 210. Also, AutG (E) surject* *s onto AutG (E0), since each coset of T 0in NG(T 0) contains an element which centrali* *zes any given element of Er E0. For any x 2 T 0of order 4 such that x2 2 E0, xgx-1 = g * *for g 2 E0 and xgx-1 = x2g for g 2 Er E0. Thus AutG (E) contains all automorphisms of E which are the identity on E0, and these form a normal subgroup of order 26. Since |O2(Aut G(E0))| 28, it now follows that |O2(Aut G(E))| 214> |CG(E)|. * *So by Proposition 8.5 again, O2(Aut L(E)) 6= 1, and E is not pivotal. Case 4: Assume E is type 2B pure of rank 4. Any such subgroup is toral [Gr2 , Theorem 9.2], and thus G-conjugate to a maximal isotropic subgroup E0 T(2). __ Any element of W which fixes an index two subgroup E T(2)must be an orthogo* *nal transvection, and hence q|_E\_E?6= 0 (see [Di, x19]). So the kernel of any root* * of G = _ _ E8(F q), when regarded as an element of Hom (T, F*q)) and restricted to T(2), i* *s also of this type, and thus does not contain any maximal isotropic subgroup of T(2). Thus CG* *(E)0 is a maximal torus by Lemma 8.3. Since the stabilizer in W (E8)=C2 ~=SO+8(2) of* * a maximal isotropic subspace is 26:L4(2) ~=26: +6(2), we also get that ß0(CG(E)) * *~=21+6+, and AutG (E) = Aut(E) ~=L4(2). Since each coset of T in NG(T ) contains elements of L, the action of AutL(E)* * on the conjugacy classes of ß0(CG(E)) is that induced by conjugation. This action of * *+6(2) on conjugacy classes in 21+6+has four orbits: the orbits of the two central ele* *ments, and those of isotropic and nonisotropic elements. Hence there are four L-conjugacy * *classes of rank four 2B-pure subgroups E, with AutL(E) ~=L4(2), 6, or 24:( 3 x 3). Si* *nce none of these automorphism groups have dihedral or semidihedral Sylow 2-subgrou* *p, E =2E 2(L ; 2) by Proposition 4.6(c). Case 5: Assume E is type 2B pure of rank 5. By [CG , Proposition 3.8], when G = E8(C), there is only one conjugacy class of such subgroups, and they have centr* *alizer of the form CG(E) ~=25+10. Hence by_[GR , Theorem A.12], there is just one conj* *ugacy class of such subgroups in G ~=E8(F q). In particular, E is G-conjugate to E0 = , where E00 T(2)is a maximal isotropic subgroup. By Case 4, AutG (E00) ~=L4(2), and since this holds for any* * index 2 subgroup of E0, we see that Aut G(E0) = Aut (E0) ~= L5(2). Also, CG(E00)=T * *~= 21+6+: generated by `, and those orthogonal transvections of T(2)which fix E00.* * Hence CG(E0) ~=T(2)o21+6+has order 215. A closer check (or a comparison with E8(C) us* *ing [GR , Theorem A.12] again) shows that Z(CG(E0)) = E0 ~=25 and CG(E0)=E0 ~=210; and also that CL(E0) = CG(E0). So under the action of NG(E0) on CG(E0) as defin* *ed in Proposition 8.5, CG(E0) acts via conjugation, hence NG(E0) leaves E0invarian* *t, and thus (since L5(2) has no subgroup of index 32) must leave a point fixed. In oth* *er words, by Proposition 8.5, for some E00which is G-conjugate to E and E0, NL(E00) = NG(* *E00). Thus by Proposition 8.5 again, AutL(E) is the stabilizer of the conjugation a* *ction of NG(E00)=CG(E00) ~=L5(2) on some conjugacy class in CG(E00) ~=25+10. Also, L5(2)* * acts on CG(E00)=E00~=210 with two nonzero orbits having stabilizers 26:( 3 x L3(2)) * *and 24: 6. The point stabilizers of elements in CG(E00)r E00thus have index 32 in* * one of the groups 26:( 3 x L3(2)) or 24: 6, while the point stabilizers of elements in* * E00are 24:L4(2) and L5(2). Hence either O2(Aut L(E)) 6= 1, or AutL(E) ~=A6, 6, or L5(* *2). In all of these cases, E =2E 2(L ; 2): either since E is not pivotal; or by Propos* *ition 4.6(c) 58 BOB OLIVER (since the Sylow 2-subgroups of 6 and L5(2) are neither dihedral nor semidihed* *ral); or by Proposition 4.6(d) (since A6 has no radical 2-subgroup of order 2). The results of this section are now summarized in the following theorem. Theorem 8.11. Fix an odd prime power q. Assume L is a simple group, isomorphic to one of the groups G2(q), 2G2(q), F4(q), 3D4(q), E6(q), 2E6(q), E7(q), or E8(* *q). Then L 2 L 2(2). 9. Sporadic groups It remains to consider the sporadic simple groups. Theorem 9.1. If L is one of the simple sporadic groups, then L 2 L 2(2). Proof.When L is one of the groups M11 , M12 , J1, or O0N , then rk2(L) 3 [GLS* *3 , x5.6]. Hence R 2(L ; 2) = ? by Proposition 4.6(b), and so L 2 L 2(2) in all of * *these cases by Proposition 4.2. The remaining sporadic groups are considered individually. We recall now (wit* *hout repeating it each time when used in the proof) that rk(E) 4. L = M22 or M23 : We have the following inclusions with odd index: M22 M23 M21:2 ~=P L3(4), where P L3(4) is the extension of P SL3(4) by the field automorphism. Hence a* *ll three of these groups have isomorphic Sylow 2-subgroups. Any elementary abelian 2-subgroup of rank 4 in P L3(4) is contained in P SL3(4), and any Sylow 2-subg* *roup of P SL3(4) contains exactly two such subgroups. Identify L as a subgroup of M24: the subgroup of elements which fix one or t* *wo points under the action on a set X of order 24. Fix S 2 Syl2(L), and let V1, V2* * S be the two elementary abelian subgroups of rank four. We take V1 to be the subg* *roup whose normalizer in M24 is the octad group V1:A8, where V1 acts freely on 16 po* *ints in X and A8 permutes the remaining 8 points in the obvious way (cf. [Gr3 , 6.8* *]). Restriction to the subgroups fixing one or two points shows that AutM22(V1) ~=A6 and AutM23(V1) ~=A7. Also, V2 is contained in O2(H) ~= 26, where H is the sextet subgroup of M24, and the O2(H)-action on X has orbits of order 4 (cf. [Gr3 , x5]). Thus V1 and V2 ar* *e not Aut(L)-conjugate, and hence are both weakly closed in S with respect to Aut(L). Since AutL (V1) has no radical subgroup of order 2, V1 =2E 2(L ; 2) by Propos* *ition 4.6(d). Thus R 2(L ; 2) V2 = ?, and hence L 2 L 2(2) by Proposition 4.2. (In fa* *ct, V2 2 R2(L ; 2) for L = M22or M23.) L = M24 or He : We refer to [A2 , x39-42] for details of the structure of thes* *e groups. In both cases, there is an involution z 2 L such that CL(z) ~=21+6+:L3(2), the * *centralizer of a transvection in L5(2), and this centralizer has odd index in L. To handle * *elements in this group, we fix V ~= (F2)5 with basis {v1, . .,.v5}, and set Vi = . We EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 59 identify H = CL(z) with the group of automorphisms of V which leave V1 and V4 invariant. For 1 i < j 5, let eij2 H be the element which sends vj 7! vi+ vj and is * *the identity on the other basis elements. Thus z = e15. Let S be the subgroup gener* *ated by all eijfor i < j; this is a Sylow 2-subgroup of H and hence of L. For each 1* * i 4, let Ui S be the subgroup of automorphisms which are the identity on Vi and on V=Vi. By [A2 , Lemma 39.1(3)] (or by the argument given below), U2 and U3 are t* *he only subgroups of S of rank six. Let E H be any elementary abelian subgroup of rank 4. For each involution u 2 E, we write K(u) = Ker(u - Id) and I(u) = Im(u - Id); thus I(u) K(u) V . In particular, I(z) = V1 and K(z) = V4. Assume first that for some u, v 2 E, I(v) K(u); i.e., (u - Id)(v - Id) 6= 0* *. Then u|I(v)and v|I(u)must be (nonidentity) involutions, which implies that dim (I(u)* *) = dim(I(v)) = 2 and dim (I(u) \ I(v)) = 1. Similarly, dim (K(u) + K(v)) = 4, so dim(K(u) \ K(v)) = 2; and the three subspaces K(u) \ K(v), I(u), I(v) are linea* *rly independent modulo I(u) \ I(v). Also, since both of these commute with z, we ha* *ve V1 K(u) \ K(v) and I(u) + I(v) V4. By an appropriate choice of basis elemen* *ts, we are reduced (up to conjugacy in H) to one of the following situations: o u = e12e35, v = e13e25, E CH () = ~=22 x D8 o u = e12e34, v = e13e24, E = CH () = ~=24 o u = e23e45, v = e24e35, E = CH () = ~=24 In the first case, for any choice of E of rank 4, Aut S(E) contains all automor* *phisms which are the identity on z and on E=, and thus contains a rank three subgro* *up. In the other two cases, AutS(E) contains all automorphisms which are the identi* *ty on the last two generators and modulo the last two generators, and thus has rank * * 4. Hence E =2E 2(L ; 2) by Proposition 4.6(c), since the Sylow subgroups of AutL(E* *) are neither dihedral nor semidihedral. T We are left with the case where W def= is contained in W 0def=u* *2EK(u). If W = W 0, then either W = V1 or V4 and E = U1 or U4, or dim (W ) = 2, 3 and (up to H-conjugacy) E Ui for i = 2, 3. A straightforward argument shows that CL(E) = CH (E) = Uiin this last case, and hence E = Uisince E is pivotal (Propo* *sition 4.4(c)). By [A2 , Lemma 40.5], Aut L(E) ~= L4(2) or 23:L3(2) when E = U1 or U4 (rk(E) = 4), and AutL(E) ~=L3(2)x 3 or 3. 6 when E = U2 or U3 (rk(E) = 6). Since none of these automorphism groups has dihedral or semidihedral Sylow 2-subgroup* *s, Proposition 4.6(c) shows that none of the Ui can lie in E 2(L ; 2). The only remaining case is where dim(W ) = 2 and dim(W 0) = 3, and hence E is H-conjugate to U2 \ U3. Thus the only subgroup of S which could lie in E 2(L ;* * 2) is U2 \ U3. It follows that R 2(L ; 2) U2U3 = ?. Since U2U3 is weakly closed * *in S with respect to Aut(L) (U2 and U3 are the only rank six subgroups of S), Propos* *ition 4.2 now implies that L 2 L 2(2). (In fact, in all three cases L = M24, L = He, * *and L = L5(2), one can show that U2 \ U3 2 E2(L ; 2) and U2U3 2 R2(L ; 2).) L = J2: This group contains two conjugacy classes of involutions, of which thos* *e of type 2A are in the centers of Sylow subgroups. By [FR1 , x3], the elements of t* *ype 2A in any elementary abelian E L form a subgroup, and there are no 2A- or 2B-pure 60 BOB OLIVER subgroups of rank 3. Thus any elementary abelian 2-subgroup E L contains an NAut(L)(E)-invariant subgroup E0 E (generated by the elements of type 2A) such that rk(E0) 2 and rk(E=E0) 2. Hence E 2(L ; 2) = ? by Proposition 4.6(c), a* *nd L 2 L 2(2) by Proposition 4.2. L = Co3 or L = HS : By [Fi, x4], there are two conjugacy classes of involutio* *ns in Co3, of which those in the center of a Sylow 2-subgroup are of type 2A with cen* *tralizer 2Sp6(2), and those of type 2B have centralizer 2xM12. By [Fi, Lemma 4.7], this * *group 2Sp6(2) has two conjugacy classes of noncentral involutions, whose centralizers* * have different orders. In other words, if x, y are commuting involutions and x has t* *ype 2A, then y and xy are conjugate in L (since their centralizers in CL(x) are isomorp* *hic), and thus have the same type. This shows that in any elementary abelian 2-subgro* *up E L, the elements of type 2A together with the identity form a subgroup of E. By [PW , Lemma 2.2 & x4], there are two conjugacy classes of involutions in * *HS, of which those in the center of a Sylow 2-subgroup have type 2A and centralizer (21+4+xC2 C4). 5, and the others have type 2B with centralizer 2 x Aut(A6). Al* *so, HS is also contained as a subgroup of Co3 (see [A2 , xx23-24]), and a compariso* *n of involution centralizers shows that the inclusion sends involutions of type 2A a* *nd 2B in HS to involutions of type 2A and 2B, respectively, in Co3. Now assume E 2 E 2(L ; 2), for L = Co3 or HS, and let E0 E be the subgroup generated by type 2A involutions. Then E0 6= 1, since E contains the center of* * a Sylow 2-subgroup (Proposition 4.4(e)), and it is clearly NAut(L)(E)-invariant. * * Thus rk(E0) 4 or rk(E=E0) 4 by Proposition 4.6(b). Since rk2(Co3) = 4 [GLS3 , p.* *305], this means that E = E0 is a rank 4 type 2A-pure subgroup. If L = Co3, then by [* *Fi, Lemma 5.9], L has a unique class of such subgroups, and AutL(E) ~=A8 ~=GL4(2) f* *or any such E. If L = HS, then by [PW , Lemma 4.1], AutL(E) ~= 6 for any such E. * *In both cases, the Sylow 2-subgroups of AutL (E) are neither dihedral nor semidihe* *dral, and hence E =2E 2(L ; 2) by Proposition 4.6(c). Thus R 2(L ; 2) = ?, and L 2 L * *2(2) by Proposition 4.2. L = McL or L = Ly : By [GLS3 , p.308], rk2(L) = 4 (see also the discussion in * *[Fi, x5] when L = McL ). By [Fi, Lemma 5.2] (when L = McL ) or [W5 , x2] (when L = Ly), AutL(E) ~=A7 for every elementary abelian 2-subgroup E L of rank 4. Since A7 contains no radical 2-subgroup of order 2, such E cannot be in E 2(L ; 2) by Pr* *oposition 4.6(d). Thus E 2(L ; 2) = ?, and hence L 2 L 2(2). L = F5 = HN : We refer to [NW , x3.1] for the following information about L. * *There are two conjugacy classes of involutions in L, types 2A and 2B. For any element* *ary abelian 2-subgroup E L, the function q: E --! F2, defined by q(v) = 1 if v is* * of type 2A and q(v) = 0 otherwise, is quadratic. Assume E 2 E 2(L ; 2). Set E0 = E \ E? (with respect to the quadratic form q), and E00= Ker(q|E0). Clearly, these subgroups are both NAut(L)(E)-invariant,* * and rk(E0=E00) 1. So either rk(E00) 4 or rk(E=E0) 4. There are two conjugacy classes of 2B-pure subgroups of rank 2 in L, and each* * such subgroup is either contained in a unique 2B-pure subgroup of rank 3, or in a un* *ique extraspecial subgroup X = 21+8+with NL(X)=X ~=(A5 x A5):2. Thus if rk(E00) 4, then NL(E) NL(E00) NL(X) (up to conjugacy), so E Z(X) ~=C2 by Proposition 4.9(a), and this contradicts the assumption rk(E00) 4. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 61 We are left with the possibility rk(E=E0) 4. Choose E0 E such that E = E0* *xE0; then q|E0 is nonsingular and rk(E0) 4. Also, E0 contains a 2A-pure subgroup * *E1 of rank two, and CL(E1) ~= 22 x A8 A12 L. Inside A12, the involutions whose support have order 4 or 12 are of type 2A, while those whose support have order* * 8 are of type 2B. Using this, one sees that either rk(E0) = 6 and CL(E0) = E0, or rk(E0) = 4 and CL(E0)_~=26 or 24x A4. Thus in all of these cases, E is containe* *d_in a unique_subgroup_E of rank 6 (on which_q_is nonsingular), and NL(E) NL(E ). Al* *so, AutL(E ) = (E , q)_~= -6(2), and so E 2= E 2(L ; 2) by Proposition 4.6(c) (Syl* *ow 2- subgroups of AutL(E ) are neither dihedral nor semidihedral). If rk(E) = 5, the* *n either q|E0 6= 0 and AutL (E) ~= 5(2) ~= 6, which again contradicts Proposition 4.6(c)* *; or q|E0 = 0 and O2(Aut L(E)) 6= 1, in which case E is not pivotal. Finally, if rk(* *E) = 4 and q is nonsingular on E, then we have just seen that the Sylow 2-subgroups of* * CL(E) are isomorphic to 26, and so E cannot be pivotal. This shows that E 2(L ; 2) = ?, and thus that L 2 L 2(2) by Proposition 4.2. In all of the remaining cases, the proof that L 2 L 2(2) will be based on a list of maximal 2-local subgroups of L _ or in some cases, a list of proper subgroup* *s of L (not necessarily 2-local) which contain all 2-local subgroups up to conjugacy* *. We label these subgroups Hn for n = 1, 2, . .,.and set Vn = Z(O2(Hn)). The goal i* *s to show that R 2(L ; 2) = ?; unless we set Q = Vn for some n, in which case we show that Q is weakly closed in some (any) Sylow 2-subgroup which contains it, and t* *hat R 2(L ; 2) Q = ?. In either case, Proposition 4.2 then implies that L 2 L 2(2). We fix a subgroup P 2 R 2(L ; 2), and set E = 1(Z(P )). If NL(E) NL(P ) * *Hn, then P O2(Hn), and so E Vn if O2(Hn) is centric in Hn. In all cases, E \ Vn* * 6= 1 by Proposition 4.9(a). If rk(Vn) = 2, then we can assume E Vn if the centrali* *zers of involutions in E appear elsewhere in the list. We use the standard notation 2A, 2B, etc. for the conjugacy classes of involu* *tions in L. Recall that by Proposition 4.4(e), E 1(Z(S)) for some S 2 Syl2(L), and* * thus E contains elements from each conjuacy class of involutions represented in Z(S). L = J3: By [FR2 , x2], L contains three conjugacy classes of maximal 2-local su* *bgroups. o NL(E) H1 ~=21+4-:A5, E V1 ~=2. Impossible since rk(V1) = 1. o NL(E) H2 ~=24:GL2(4) , E V2 ~=24. Then E = V4 and AutL (E) ~=GL2(4) ~= C3 x A5. Since A5 has no radical subgroup of order 2, this contradicts Proposi* *tion 4.6(d). o NL(E) H3 ~=22+4:(3 x 3), E V3 ~=22. Impossible since rk(V3) = 2. L = Suz : We refer to [W2 ] for the following information. There are two conju* *gacy classes of elements of order 2 in L, of which those of type 2A are in the cente* *rs of Sylow 2-subgroups. Also, in any elementary abelian subgroup E L, the involuti* *ons of type 2A together with the identity form a subgroup of E. Hence NL(E) is cont* *ained in the normalizer of some 2A-pure subgroup; and hence by [W2 , x2.4], in one of* * the groups Hn in the following list. o NL(E) H1 ~=21+6-.U4(2), E V1 ~=2. Impossible since rk(V1) = 1. 62 BOB OLIVER o NL(E) H2 ~=22+8:(A5 x 3), E V2 ~=22. Impossible since rk(V2) = 2. o NL(E) H3 ~=24+6:3A6 , E V3 ~=24. Then E = V3 and AutL (E) ~=A6, which contradicts Proposition 4.6(d) (A6 contains no radical 2-subgroup of order 2). L = Ru : There are two conjugacy classes of involutions, of which those of type* * 2A lie in the centers of Sylow 2-subgroups. Thus all pivotal 2-subgroups of L con* *tain elements of type 2A (Proposition 4.4(e)). By [W3 , x2.4], the involutions of ty* *pe 2A in E together with the identity form a subgroup E0 E, and thus NL(E) NL(E0). By [W3 , x2.4-2.5], the normalizer of each 2A-pure elementary abelian subgrou* *p of L is conjugate to a subgroup of one of the subgroups Hn listed below. In all ca* *ses, Vi is 2A-pure, and thus E = E0. For V3 ~=26 this is shown in [W3 , Lemma 1] (where* * V3 is denoted R1). o NL(E) H1 ~=2.24+6: 5, E V1. From the description in [A1 , 12.12], we get * *that V1 = Z(O2(H1)) has rank 1. Alternatively, this follows directly from the commu* *tator relations listed in [P2 , Lemma 12] (where O2(H1) is the subgroup generated by* * the eleven elements z, t, v, w, w1, x1, x2, a, b, c, d). Thus this case is impossi* *ble. o NL(E) H2 ~=23+8:L3(2), E V2 ~=23. Impossible since rk(V2) = 3. o NL(E) H3 ~=26:G2(2) , E V3 ~=26. Assume here that NL(E) is not conjugate * *to a subgroup of H1 or H2. Let T be the conjugacy class of the subgroups of rank * *2 in V2. Let D be the "diagramö f E in the sense of [W3 ]: the graph with one node for* * each involution in E, and an edge connecting two nodes whenever the elements genera* *te a subgroup in T . The automorphism group AutL(E) acts on D (and Out(L) = 1 in th* *is case). There cannot be any AutL(E)-invariant node or triangle in D, since this* * would imply an AutL (E)-invariant element or subgroup in T , hence that NL(E) H1 or H2, contradicting our assumption on E. By [W3 , x2-4-2.5], for each E V3 of rank 2, either the diagram D contains* * an Aut L(E)-invariant node or triangle; or D is a disjoint union of two or more t* *riangles and isolated nodes in which case CL(E) = V3 ~= 26; or E = V3 and Aut L(E) ~= G2(2) ~=U3(3):2; or rk(E) = 2 and NL(E)=V3 ~=( 3 x 3). Since rk(E) 4, this shows that CL(E) ~= V3, and hence (since E is pivotal) that E is conjugate to * *V3. Thus (E, AutL(E)) ~=(26, G2(2)), and this contradicts Proposition 6.4. L = F3 = Th : By [W7 , Theorem 2.2], each 2-local subgroup of L is conjugate t* *o a subgroup of H1 or H2 as listed here. o NL(E) H1 ~=21+8+.A9, E V1 ~=2. Impossible since rk(V1) = 1. o NL(E) H2 ~=25.L5(2), E V2 ~=25. Then AutL(E) = Aut(E), which contradicts Proposition 4.7(a). L = J4: By [KW , x2], there are four conjugacy classes of maximal 2-local subg* *roups, as listed below. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 63 __def o NL(E) H1 ~=210:L5(2), E V1 ~=210. If NL(E) H1 and E V1, then_P_ =_ P=V1 is a radical 2-subgroup of H1=V1 ~=L5(2), and NL(P )=P ~=NL5(2)(P )=P . A* *lso, L5(2) acts irreducibly on V1 [Ja, Theorem A], and hence acts as 2(V ), where * *V is one of the standard 5-dimensional representations. The only radical subgroups * *which have fixed subspace on V1 of rank 4 are the trivial subgroup, and two subgro* *ups 24 with normalizer 24:L4(2) with fixed subspaces of rank 4 or 6. By Proposition 4* *.6(c), no such subgroup can be in R 2(L ; 2). o NL(E) H2 ~=23+12.( 5 x L3(2)), E V2 ~=23. Impossible since rk(V2) = 3. o NL(E) H3 ~=21+12+.(3M22:2), E V3 ~=2. Impossible since rk(V3) = 1. o NL(E) H4 ~=211:M24 , E Q = V4 ~=211 . By [KW , Lemma 1.1.2], Q = V4 is the unique subgroup of NL(Q) ~=211:M24of rank 11; and this group contains a Sy* *low 2-subgroup S 2 Syl2(L). Hence Q = V4 is weakly closed in S with respect to Aut* *(L); and we have just shown that R 2(L ; 2) Q = ?. L = Co1 : By [Cu , Theorem 2.1], the normalizer of each elementary abelian 2-su* *bgroup E L is contained in one of the seven subgroups listed below. We set Q = V2, a* *nd show that R 2(L ; 2) Q = ?. o NL(E) H1 ~=21+8+. +8(2), E V1 ~=2. Impossible since rk(V1) = 1. o NL(E) H2 ~=24+12.(3S6 x 3), E V2 ~=24. Then Aut L(E) = Sp4(2) ~= 6, and this contradicts Proposition 4.6(c) (Sylow subgroups are neither dihedral * *nor semidihedral). o NL(E) H3 ~=22+12.( 3 x A8), E V3 ~=22. Impossible since rk(V3) = 2. o NL(E) H4 ~=Co2 . Upon examination of the proof of [Cu , Theorem 2.1], one s* *ees that the only case where NL(E) is not contained in one of the other subgroups * *Hn occurs when E = <(4), oe> (in the notation of p.419) is a certain "BD-pure" su* *bgroup (all involutions have type 2A in the notation of the atlas) of rank 4 (rank 5 * *in 2Co1). As noted by Wilson in [W1 , p.112], this subgroup is contained in a unique ran* *k 5 subgroup <(7), oe> whose involutions all have type 2A, and hence its normalize* *r is contained in the normalizer of that subgroup, which is contained in some subgr* *oup conjugate to H1. We can thus ignore the case NL(E) Co2. o NL(E) H5 ~=(A4 x G2(4)).2 . Then by Lemma 1.12(a,b), P \ (A4 x G2(4)) = P1xP2 where P1 ~=22 A4 and P2 G2(4) are radical 2-subgroups. By examination of the two maximal parabolic subgroups 22+8:(3 x A5) and 24+6:(A5x 3) of G2(4)* *, we see that the only possibility for E with an irreducible component of rank 4 * *(see Proposition 4.6(b)) is E = 22 x 24 A4 x G2(4). In this case, E lifts to a subgroup of 2Co1 isomorphic to Q8x 24; NL(E) is con* *tained in the normalizer of the second factor, whose elements are not of type 2B (sin* *ce they lift to involutions in 2Co1); and hence by [Cu , Lemmas 2.2 & 2.5], this norma* *lizer is contained in one of the subgroups Hn for n = 1, 2, 3, 4 above. 64 BOB OLIVER o NL(E) H6 ~=(A6 x U3(3)).2. Since rk2(A6) = rk2(U3(3)) = 2, E has a filtrati* *on by NL(E)-invariant subgroups for which the quotients all have rank 2, and th* *is contradicts Proposition 4.6(b). o NL(E) H7 ~=211.M24 , E Q = V7 ~=211 . By [A1 , (30.3) & (31.11)], Q is the unique subgroup of H7 of rank 11, and hence weakly closed (with respect to Aut* *(L)) in any Sylow 2-subgroup which contains it. We have now shown that R 2(L ; 2) Q = * *?. L = Co2 : By [W1 , x3], the normalizer of each elementary abelian 2-subgroup of* * L is contained in one of the subgroups Hn in the following list. o NL(E) H1 = 24+10.( 5 x 3), E V1 ~=24. Then E = V4 and AutL (E) ~= 5, and an examination of the first diagram in [W1 , p.113] shows that AutL(E) act* *s via the permutation representation (with two orbits of lengths 5 and 10). This con* *tradicts Proposition 4.7(b). o NL(E) H2 = 21+8+:Sp6(2), E V2 ~=2. Impossible since rk(V2) = 1. o NL(E) H3 = 21+6+.24A8, E V3 ~=2. Impossible since rk(V3) = 1. o NL(E) H4 = M23 . This subgroup arises as (one possible) intersection of a s* *ub- group 211.M24 Co1 with Co2. From the analysis in [Cu , x2], we see that each * *time the normalizer of an elementary abelian subgroup E Co1 was shown to be con- tained in a subgroup K ~=211:M24, it was contained in such a way that E inters* *ects nontrivially with the rank 11 subgroup. Hence if NL(E) Co2, then K \ Co2 can* *not be isomorphic to M23, and so we can ignore this case. o NL(E) H5 = U6(2):2 . Then P 2 R 2(H5; 2) by Proposition 4.9(b), which is empty by Lemma 6.5. (Note that Out(L) = 1.) o NL(E) H6 = McL . Then rk(E) = rk2(McL ) = 4, and Aut L(E) ~= A7 as de- scribed above. This is impossible by Proposition 4.6(d), since A7 contains no * *radical subgroups of order 2. o NL(E) H7 = 210:M22:2 , E Q = V7 ~=210 . By [A1 , (30.3) & (31.11)], V7 is the unique rank 10 subgroup of H7, and hence weakly closed in any Sylow subgro* *up which contains it. We have just shown that R 2(L ; 2) Q = ?. L = Fi22: By [A3 , (25.7)], for any S 2 Syl2(L), the set of involutions in S of* * type 2A generates a subgroup 210, which thus is weakly closed in S with respect to Aut(* *L). We fix S, and let Q ~= 210 denote this subgroup. We will show that R 2(L ; 2) Q =* * ?, and also that R 2(Aut (L) ; 2) Q = ?. The latter will be needed later, when wor* *king with the group F i024. We set = Aut(L) = F i22:2 for short [A3 , (37.2)]. Thr* *oughout the following discussion, we use the term "transposition" to refer to involutio* *ns of type 2A; these all have the property that the product of any two of them has order 2* * or 3. By [W4 , Proposition 4.4]_or [Fl], for each_elementary abelian 2-subgroup 1 6* *= E L, NL(E) Hn and N (E) H nfor some Hn and H n(n = 1, . .,.5) as described in the following list: EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 65 __ o NL(E) H1 =_2U6(2) , N (E) H 1 = 2U6(2).2, E V1 ~=2. If NL(E) H1 and N (E) H 1, then P=V1 is in R 2(U6(2) ; 2) or R 2(U6(2):2 ; 2) by Proposi* *tion 4.9(b); and these sets are empty by Lemma 6.5. __ o NL(E) H2 = (2 x 21+8+:U4(2)):2, N (E) H 2= (2 x 21+8+:U4(2):2):2, E V2 * *~= 22. Impossible since rk(V2) = 2. __ o NL(E) H3 = 25+8:( 3 x A6), N (E) H 3= 25+8:( 3x 6), E V3 ~=25. Thus Aut L(E) is the stabilizer of the action of A6 or 6. As described in [W4 , x* *x3-4], V3 ~=25 is generated by a unique hexad of transpositions, and hence AutL(V3) ~* *=A6 and Aut (V3) ~= 6 act on V3 via the permutation action on F62=F2. Since neithe* *r A6 nor 6 has a radical subgroup of order 2, Proposition 4.5 shows that E cannot * *be equal to V3, nor equal to the index two subgroup of V3 containing no transpositions * *(since that is also stabilized by A6). The only remaining possibility is for E to hav* *e rank 4, containing exactly three or four transpositions (if it only has one or two its* * normalizer is contained in H1 or H2). In these cases, either AutL(E)=O2(Aut L(E)) ~= 3 (t* *hree transpositions) or Aut L(E) = Aut (E) ~= 4 (four transpositions); and in nei* *ther case can E be contained in E 2(L ; 2) (Proposition 4.6). __ o NL(E) H4 = 26:Sp6(2) , N (E) H 4 = 27:Sp6(2), E V4 ~= 26. By [W4 , xx3-4], the only situation where NL(E) H4 and is not contained in any of the other subgroups Hn occurs when all involutions in E have type 2B (in particula* *r, E contains no transpositions), and E supports a nonsingular symplectic form b su* *ch that Aut L(E) = Aut(E, b). Thus by Proposition 6.4 (assuming rk(E) 4), E is in ne* *ither E 2(L ; 2) nor E 2( ; 2). Similarly, if E and_E0 def=E\L is 2B-pure and_s* *upports a symplectic form as described in (e), then E H 4~=27:Sp6(2), so E O2(H 4)* * by Proposition 4.9(a), hence either Aut (E) ~=Aut L(E0) or O2(Aut (E)) 6= 1; and* * the same reasoning shows that E =2E 2( ; 2). __ o NL(E) H5 = 210:M22 , N (E) H 5= 210:M22:2, E Q = V5 ~=210 . We have already seen that Q is weakly closed in any Sylow 2-subgroup which contains it* *; and we have just shown that R 2(L ; 2) Q = ?. L = Fi23: Note that Out(L) = 1 [A3 , (37.2)]. Fix S 2 Syl2(L). Then Z(S) ~=22, * *and contains a representative from each of the three classes of involutions in L. T* *hus by Proposition 4.4(e), each critical 2-subgroup of L contains involutions from eac* *h of the three classes. By [A3 , (25.7)], the set of transpositions in S (involutions of* * type 2A) generates a subgroup 211, which thus is weakly closed in S. We let Q ~=211denot* *e this subgroup, and will show that R 2(L ; 2) Q = ?. By [Fl, x2], the normalizer of each elementary abelian 2-subgroup of L which * *contains transpositions is contained in one of the subgroups in the following list. o NL(E) H1 ~=2Fi22, E V1 ~=2. By Proposition 4.9(b) (and since Out(L) = 1), P=V1 2 R 2(F i22; 2). We have already seen that this implies that P=V1 contain* *s the subgroup 210 generated by the involutions of type 2A in some Sylow 2-subgroup * *of F i22, and hence that P Q = V5 (up to conjugacy). 66 BOB OLIVER o NL(E) H2 ~=22.U6(2).2 , E V2 ~=22. Since Out (L) = 1, Proposition 4.9(b) implies that P=V2 2 R 2(U6(2):2 ; 2), and this set is empty by Lemma 6.5. o NL(E) H3 ~=(22 x 21+8+).(3 x U4(2)).2, E V3 ~=23. Impossible since rk(V3)* * = 3. o NL(E) H4 ~=26+8:( 3 x A7) , E V4 ~=26. The subgroup V4 contains (and is generated by) exactly seven transpositions, which are permuted in the obvious * *way by AutL(V4) ~=A7. Hence AutL(E) is the stabilizer of this permutation action o* *f A7. Since A7 has no radical 2-subgroup of order 2, V4 =2E 2(L ; 2) by Proposition * *4.6(d)), and thus E V4. Thus P=O2(H4) ~=P1 x P2, where P1 3 and 1 6= P2 A7 are radical 2-subgroups (Lemma 1.12(a)); E is the fixed subgroup of the P2-action * *on V4; and this is impossible since the nontrivial radical 2-subgroups of A7 all * *have fixed subgroup on V4 of rank 3. o NL(E) H5 ~=211.M23 , E Q = V5 ~=211 . We have already seen that Q is weakly closed in any Sylow 2-subgroup which contains it; and we have just shown that R 2(L ; 2) Q = ?. L = Fi024: By [A3 , (37.1)], Out (L) ~=C2, and Aut(L) = F i24. We write = F i* *24 for short. The group is generated by transpositions: elements in a conjugacy cla* *ss of involutions in r L the product of any two of which has order 2 or 3. By [A3 , * *(37.4)], L has two conjugacy classes of involutions: each element of type 2A is a produc* *t of a unique pair of commuting transpositions (its factors), while each element of ty* *pe 2B is a product of four commuting transpositions (but not uniquely). Fix bS2 Syl2( ), and set S = bS\ L 2 Syl2(L). By [A3 , (25.7)], the set of tr* *ansposi- tions in bSgenerates a subgroup bQ~=212. Set Q = bQ\L ~=211. By [A3 , (34.9)], * *Q is the Todd module for NL(Q)=Q ~=M24. Hence by [A1 , 31.11], Q is the unique elementary abelian 2-subgroup of NL(Q) of rank 11. Since Q C S, NL(Q) S, and thus Q is weakly closed in S with respect to Aut(L). We will show that R 2(L ; 2) Q = ?. By [W6 , Theorems D & E] (with corrections in [LW , x2]), each 2-local subgro* *up of L is contained up to conjugacy in one of the subgroups in the following list. o NL(E) H1 = N(2A ) ~=2Fi22:2, E V1 ~=2. Then N (E) N (H1), since the factors of the generator of V1 normalize E. By Proposition 4.9(b) again, P=V1* * 2 R 2(F i22:2 ; 2). We have already shown that this implies that P=V1 contains (* *up to conjugacy) the subgroup 210 generated by the involutions of type 2A in any Syl* *ow 2-subgroup of F i22. Since all involutions in F i22 lift to involutions in 2F* * i22 [A3 , (23.8)], this shows that P contains a subgroup 211, which must be conjugate to* * Q. o NL(E) H2 = N(2B ) ~=21+12+.3U4(3):2, E V2 ~=2. Impossible since rk(V2) = * *1. o NL(E) H3 ~=22.U6(2): 3 , E V3 ~=22. In this case, V3 Q (up to conjugacy* *), and all of its involutions are of type 2A. The factors of the involutions in V* *3 all lie in bQ, and generate a rank three subgroup with just three transpositions, whic* *h are permuted by the conjugation action of E and hence normalize E. Thus N (E) N (H3), and so P=V3 2 R 2(U6(2): 3; 2) by Proposition 4.9(b). But this set is * *empty by Lemma 6.5. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 67 o NL(E) H4 ~=26+8:( 3 x A8) , E V4 ~= 26. Here, Aut L(V4) acts on V4 via the natural action of A8 ~= +6(2). This follows from the discussion in [W6 , * *p.91], where it is shown that V4 is the intersection with F i024of the subgroup of ra* *nk 7 in F i24generated by the eight transpositions in an octad. Hence E V4 by Propos* *ition 4.6(c), since the Sylow 2-subgroups of A8 are neither dihedral nor semidihedra* *l. Hence P=V4 is a nontrivial radical 2-subgroup of A8, and E is the fixed subgroup if * *its action on V4. But the fixed subgroup of any such radical 2-subgroup has rank 3, and* * so this case is not possible. o NL(E) H5 ~=23+12.(A6 x L3(2)), E V5 ~=23. Impossible since rk(V5) = 3. o NL(E) H6 ~=(A4 x +8(2):3):2, E V6 ~=22. Then P A4 x +8(2), and is a radical 2-subgroup of this product. So by Lemma 1.12(b), P = V6 x P 0, where P 0is a radical subgroup of +8(2), and hence of the form O2(P) for some parab* *olic subgroup P +8(2) for which rk(Z(O2(P))) 4. The only such subgroup is of t* *he form P ~=26: +6(2), which would imply that (E, AutL(E)) ~=(22 x 26, (3 x +6(2)):2). By Proposition 6.4, no such group E can be in E 2(L ; 2). o NL(E) H7 ~=28: -8(2), E V7 ~= 28. The maximal parabolic subgroups of -8(2) are P1 ~= 26:U4(2) (O2(P1) has 1-dimensional fixed subgroup on V7 ~= 28* *), P2 ~= 21+8+:( 3 x A5) (2-dimensional fixed subgroup), and P3 ~= 23+6:(L3(2) x * *3) (3-dimensional fixed subgroup). Hence P V7 would imply rk(E) < 4; while Prop* *o- sition 6.4 shows that V7 =2E 2(L ; 2). o NL(E) H8 ~=211.M24 , E Q = V8 ~=211 . We have already seen that Q is weakly closed in S with respect to Aut(L); and have now shown that R 2(L ; 2) * *Q = ?. L = F2: This group contains four conjugacy classes of involutions. By [Gr1 , Th* *eorem 3], or by [Sg, Theorem 5.6], Out (L) = 1. By [MS ], every 2-local subgroup of * *L is contained in one of the subgroups in the following list. o NL(E) H1 = N(2B ) ~=21+22.Co2 , E V1 ~=2. Impossible since rk(V1) = 1. o NL(E) H2 ~=22+10+20(M22:2 x 3) , E V2 ~=22. Impossible since rk(V2) = 2. o NL(E) H3 ~=[235]( 5 x L3(2)), E V3 ~=23. Thus rk(E) 3. The subgroup H3 is described in [A1 , p.33], and its construction in [A1 , pp.219-220]. In * *particular, H3 = NL(W ) for a certain subgroup W O2(H1) ~= 21+22 of rank 3 (and with Z(H1) W ), and H3=CL(W ) ~=L3(2). Since H1 is the centralizer of an element * *of W , this shows that CO2(H1)(W ) C O2(H3) H1. Since O2(H1) is centric in H1, * *this shows that Z(O2(H3)) CO2(H1)(W ), and hence (since O2(H1) is extraspecial) t* *hat W = V3. o NL(E) H4 ~=25+5+10+10L5(2) , E V4 ~= 25. Then Aut L(E) = Aut (E), and this contradicts Proposition 4.6(b,c) (the Sylow 2-subgroups of AutL(E) are ne* *ither dihedral nor semidihedral). 68 BOB OLIVER o NL(E) H5 ~=29+16.Sp8(2), E V5 ~=29. Then E V5 ~=29, and P=O2(H5) is a 2-radical subgroup of H5=O2(H5) ~=Sp8(2). The only radical 2-subgroups of Sp8(* *2) whose fixed subgroup on 28 has rank 4 are the trivial subgroup (and (Aut L(E* *), E) ~= (Sp8(2), 29) implies E =2E 2(L ; 2) by Proposition 6.4); and 210:A8 ~=210:L4(2* *) (im- possible by Proposition 4.6(c), since the Sylow subgroups are neither dihedral* * nor semidihedral). o NL(E) H6 = N(2A ) ~=2.2E6(2):2. Then |V6| = 2, and P=V6 2 R 2(H6=V6; 2) by Proposition 4.9(b). In particular, (P=V6)\2E6(2) is a radical 2-subgroup of* * 2E6(2). From the list of maximal parabolic subgroups of 2E6(2), we see that the only r* *adical 2- subgroup whose center has rank 4 has the form 28+16, with normalizer 28+16:S* *O-8(2) in 2E6(2):2. Hence the only possibilities are (E=V6, AutL(E=V6)) ~=(28, SO-8(2)) or (27, Sp6(2)). (The second case corresponds to the extension of 28+16by a transvection in SO-* *8(2).) By Proposition 6.4, neither of this situations can occur for P=V6 2 R 2(2E6(2)* *:2 ; 2), and so this case is impossible. o NL(E) H7 = N(2C ) ~=(22 x F4(2)):2. From the list of maximal parabolic sub- groups of F4(2), we see that only two radical 2-subgroups U = O2(P) have cente* *rs of rank 4. In both cases, this would mean P = U of order 220, E = Z(P ) ~=25, a* *nd Aut L(E) ~=P=U ~= 3 x L3(2); with a rank 2 subgroup of E0 E which is normal in P. Which contradicts Proposition 4.6(b). o NL(E) H8 = N(2C 2) ~= 4 x 2F4(2) . The radical 2-subgroups of 2F4(2) have centers of rank one or two (see [W3 ] or [P1 ]), and hence E =2 E 2(L ; 2) and* * P 2= R 2(L ; 2). L = F1: By [Gr1 , Theorem 3], or by [GMS , Theorem 5.10], Out (L) = 1. By [MS* * ], every 2-local subgroup of L is contained in one of the subgroups in the followi* *ng list. o NL(E) H1 = N(2A ) ~=2.F2 , E V1 ~= 2. By Proposition 4.9(b), P=V1 2 R 2(F2; 2), and we have already shown that this last set is empty. o NL(E) H2 = N(2A 2) ~=22.2E6(2): 3 , E V2 ~=22. Then (P=V2) \ 2E6(2) is a radical 2-subgroup of 2E6(2). All involutions in 2E6(2) lift to involutions in* * its 2-fold cover, so any elementary abelian 2-subgroup lifts to an elementary abelian 2-s* *ubgroup in the 2-fold cover. From the list of maximal parabolic subgroups of 2E6(2), w* *e see that the only radical 2-subgroup whose center has rank 4 has the form 28+16,* * with normalizer 28+16: -8(2). Also, the only involutions z 2 SO-8(2)r -8(2) such * *that is radical in SO-8(2) are the orthogonal transvections, with C -8(2)(z) ~=* *Sp6(2). Hence either P=V2 ~=28+16, E=V2 ~=28, and AutL(E=V2) ~= -8(2); or P=V2 ~=28+16* *.2, E=V2 ~=27, and AutL (E=V2) ~=Sp6(2). Both of these cases contradict Propositi* *on 6.4. o NL(E) H3 = N(2B ) ~=21+24.Co1 , E V3 ~=2. Impossible since rk(V3) = 1. o NL(E) H4 ~=22+11+22(M24x 3) , E V4 ~=22. Impossible since rk(V4) = 2. EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 69 o NL(E) H5 ~=23+6+12+18.(L3(2) x 3S6), E V5 ~=23. Impossible since rk(V5) = 3. o NL(E) H6 ~=25+10+20.(L5(2) x 3), E V6 ~=25. Then Aut L(E) = Aut (E), which contradicts Proposition 4.7(a). o NL(E) H7 ~=210+16. +10(2), E V7 ~=210. Then P O2(H7), and P=O2(H7) is a 2-radical subgroup of H7=O2(H7) ~= +10(2). By examination of the root sys* *tem of the Lie group D5, one sees that the only radical 2-subgroups of +10(2) who* *se fixed subgroup on 210 has rank 4 are the trivial subgroup (and (Aut L(E), E)* * ~= ( +10(2), 210) implies E 2= E 2(L ; 2) by Proposition 6.4); two conjugacy clas* *ses of subgroups 210 with normalizers 210:L5(2) (5-dimensional fixed subspace), and o* *ne conjugacy class of subgroup 26+8 with normalizer 26+8:L4(2) (4-dimensional fix* *ed subspace). Hence in both of these cases, Aut L(E) = Aut (E), which implies E * *2= E 2(L ; 2) by Proposition 4.7(a). This finishes the proof of Theorem 9.1. 10. Computations of lim1(ZG) In this section, we summarize what we know about the groups lim-1(YeL), with * *in- dications of the proof in some cases. In all cases, lim-0(YeL) = 0 by Glauberm* *an's Z*-theorem [Gl]. Using this, the following proposition follows by essentially t* *he same proof as that of Proposition 4.2. Proposition 10.1. Fix a finite simple group L and S 2 Syl2(L). Assume Q S is * *2- centric in L and weakly closed in S with respect to Aut(L), and that it has the* * property that R 2(L ; p) Q = ?, and that no subgroup in R1(L ; p) Q is contained in any * *other subgroup in this set. Let eLbe a quasisimple group such that Z(eL) is a 2-grou* *p and eL=Z(eL) ~=L, and let Aut(eL) be a subgroup which contains Inn(eL) ~=L. The* *n for any set P1, . .,.Pk of -conjugacy class representatives for subgroups in R1(L * *; p) Q , hMk i lim-1(YeL) ~=Ker 1(N (Pi), NL(Pi); YeL(Pi)) --- i H0(N (Q); YeL(Q)) i=1 for some surjection between these two groups. When L ~=An, then ( Z=2 if eL~=An, n, n 2, 3 (mod 4) lim-1(YeL) ~= O2( ) 0 otherwise; When n, this is shown via an easy modification of the proof of Theorem 5.1. (The case L = A6 ~=P SL2(9) and 6 must be handled separately.) Recall that we write E2k n for the elementary abelian group 2k acting with one free orbit* *, and that Q An denotes the product of [n=4] copies of E4. Then R1(An ; 2) Q is emp* *ty i * * j if n 0, 1 (mod 4); and contains the conjugacy class of the group An \ S0x Ex* *32 for some S0 2 Syl2( n-6) if n 2, 3 (mod 4). So An 2 L1(2) in the first case * *by 70 BOB OLIVER Proposition 4.2, and one gets the above computations using Proposition 10.1 in * *the second case. Note, however, that lim-1(Z n) = 0 for all n, even in those cases where lim-1* *(YAnn) ~= Z=2. This follows from the observation that lim-0(Z n=YAnn) ~=Z=2 when n 2, 3* * (mod 4). When L is of Lie type in characteristic two, then by Theorem 6.3, L 2 L1(2) e* *xcept when L = L3(2) ~= L2(7). When L is of Lie type in odd characteristic, and not isomorphic to E7(q) or E8(q), then L 2 L1(2) except for the following cases: o lim-1(YL) ~=Z=2 if L ~=P SL2(q) ~= 3(q), q 1 (mod 8), Autfg(L) o lim-1(YL) ~=Z=2 if L ~=P SL4(q), q 3 (mod 4), Autfg(L) o lim-1(YL) ~=Z=2 if L ~=P SU4(q), q 1 (mod 4), Autfg(L) o lim-1(YL) ~=Z=2 if L ~= ns2n(q), n 3, Autfg(L). Here, Autfg(L) denotes the group generated by inner, field, and graph automorph* *isms of L. Also, ns2n(q) = (F2nq, q) where q is a quadratic form with nonsquare di* *scriminant. The case L = P SL2(q) is shown using Proposition 1.6 (or an easy modification of the proposition and its proof). In all other cases, there is a weakly closed s* *ubgroup Q such that R1(L ; 2) Q contains at most one conjugacy class, or in certain ca* *ses two L-conjugacy classes which are Aut (L)-conjugate. The results then follow f* *rom Propositions 4.2 and 10.1. The sporadic groups all lie in L1(2), with the possible exception of the baby* * monster and the monster. This is shown, either by modifying and extending the arguments used in Section 9, or by using lists of radical 2-subgroups of these groups suc* *h as those published by Yoshiara [Y1 ] [Y2 ] and by An & O'Brien [AO ]. References [AO] J. An & E. O'Brien, The Alperin and Dade conjectures for the O'Nan and Rud* *valis simple groups, Comm. Algebra 30 (2002), 1305-1348 [A1] M. Aschbacher, Overgroups of Sylow subgroups in sporadic groups, Memoirs A* *mer. Math. Soc. 343 (1986) [A2] M. Aschbacher, Sporadic groups, Cambridge Univ. Press (1994) [A3] M. Aschbacher, 3-Transposition groups, Cambridge Univ. Press (1997) [BL] C. Broto, R. Levi, On spaces of self homotopy equivalences of p-completed * *classifying spaces of finite groups and homotopy group extensions, Topology 41 (2002), 229-255 [BLO]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed class* *ifying spaces of finite groups, Invent. math. 151 (2003), 611-664 [Bb] N. Bourbaki, Groupes et alg`ebres de Lie, Chapitres 4-6 [BW] N. Burgoyne & C. Williamson, On a theorem of Borel and Tits for finite Che* *valley groups, Arch. Math. Basel 27 (1976), 489-491 [Ca1]R. Carter, Simple groups of Lie type, Wiley (1972) [Ca2]R. Carter, Finite groups of Lie type: conjugacy classes and complex charac* *ters, Wiley (1985) [CG] A. Cohen & R. Griess, On finite simple subgroups of the complex Lie group * *of type E8, Proc. Symp. Pure Math. 47 (1987), 367-405 [Cu] R. Curtis, On subgroups of .O II. Local structure, J. Algebra 63 (1980), 4* *13-434 [Di] J. Dieudonn'e, Sur les groupes classiques, Hermann (1967) EQUIVALENCES OF CLASSIFYING SPACES COMPLETED AT THE PRIME TWO 71 [Fi] L. Finkelstein, The maximal subgrops of Conway's group C3 and McLaughlin's* * group, J. Algebra 25 (1973), 58-89 [FR1]L. Finkelstein & A. Rudvalis, Maximal subgroups of the Hall-Janko-Wales gr* *oup, J. Algebra 24 (1973), 486-493 [FR2]L. Finkelstein & A. Rudvalis, The maximal subgroups of Janko's simple grou* *p of order 50, 232, 960, J. Algebra 30 (1974), 122-143 [Fl] D. Flaass, 2-local subgroups of Fischer groups, Math. Zametki 35 (1984), 3* *33-342 [Gl] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), * *403-420 [Gor]D. Gorenstein, Finite groups, Harper & Row (1968) [GL] D. Gorenstein & R. Lyons, The local structure of finite groups of characte* *ristic 2 type, Memoirs Amer. Math. Soc. 276 (1983) [GLS3]D. Gorenstein, R. Lyons, & R. Solomon, The classification of the finite s* *imple groups, nr. 3, Amer. Math. Soc. surveys and monogr. 40 #3 (1997) [Gr1]R. Griess, The structure of the monster simple group, Proc. Conf. on Finit* *e Groups (Park City 1975), Academic Press (1976), 113-118 [Gr2]R. Griess, Elementary abelian p-subgroups of algebraic groups, Geometriae * *Dedicata 39 (1991), 253-305 [Gr3]R. Griess, Twelve sporadic groups, Springer-Verlag (1998) [GMS] A uniqueness proof for the monster, Annals of Math. 130 (1989), 567-602 [GR] R. Griess & A. Ryba, Embeddings of PGL2(31) and SL2(32) in E8(C), Duke Mat* *h. J. 94 (1998), 181-211 [Gro]J. Grodal, Higher limits via subgroup complexes, Annals of Math. 155 (2002* *), 405-457 [Hum]J. Humphreys, Linear algebraic groups, Springer-Verlag (1975) [Hp] B. Huppert, Endliche Gruppen I, Springer-Verlag (1967) [HB3]B. Huppert & N. Blackburn, Finite groups III, Springer-Verlag, Berlin (198* *2) [JM] S. Jackowski & J. McClure, Homotopy decomposition of classifying spaces vi* *a elementary abelian subgroups, Topology 31 (1992), 113-132 [JMO]S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-map* *s of BG via G- actions, Annals of Math. 135 (1992), 184-270 [Ja] Z. Janko, A new finite simple group of order 86,775,571,046,077,562,880 wh* *ich possesses M24 and the full cover of M22as subgroups, J. Algebra 42 (1976), 564-596 [KW] P. Kleidman & R. Wilson, The maximal subgroups of J4, Proc. London Math. S* *oc. 56 (1988), 484-510 [LW] S. Linton & R. Wilson, The maximal subgroups of the Fischer groups Fi24and* * Fi024, Proc. London Math. Soc. 63 (1991), 113-164 [McL]S. Mac Lane, Categories for the working mathematician, Springer-Verlag (19* *71) [MP] J. Martino & S. Priddy, Unstable homotopy classification of BG^p, Math. Pr* *oc. Cambridge Phil. Soc. 119 (1996), 119-137 [MS] U. Meierfrankenfeld & S. Shpektorov, Maximal 2-local subgroups of the mons* *ter and baby monster, manuscript [NW] S. Norton & R. Wilson, Maximal subgroups of the Harada-Norton group, J. Al* *gebra 103 (1986), 362-376 [Ol] B. Oliver, Equivalences of classifying spaces completed at odd primes, Mat* *h. Proc. Cambridge Phil. Soc. (to appear) [P1] D. Parrott, A characterization of the Tits' simple group, Canad. J. Math. * *24 (1972), 672-685 [P2] D. Parrott, A characterization of the Rudvalis simple group, Proc. London * *Math. Soc. 32 (1976), 25-51 [PW] D. Parrott & S. Wong, On the Higman-Sims simple group of order 44,352,000,* * Pacific J. Math. 32 (1970), 501-516 [Sg] On the uniqueness of Fischer's baby monster, Proc. London Math. Soc. 62 (1* *991), 509-536 [Se] J.-P. Serre, Linear representations of finite groups, Springer-Verlag (197* *7) [Sh] K. Shinoda, A characterization of odd order extensions of the Ree groups 2* *F4(q), J. Fac. Sci. Univ. Tokyo 22 (1975), 79-102 [St] R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs Amer. Math* *. Soc. 80 (1968) [Sz] M. Suzuki, Group theory I, Springer-Verlag (1982) [W1] R. Wilson, The maximal subgroups of Conway's group .2, J. Algebra 84 (1983* *), 107-114 72 BOB OLIVER [W2] R. Wilson, The complex Leech lattice and maximal subgroups of the Suzuki g* *roup, J. Algebra 84 (1983), 151-188 [W3] R. Wilson, The geometry and maximal subgroups of the simple groups of A. R* *udvalis and J. Tits, Proc. London Math. Soc. 48 (1984), 533-563 [W4] R. Wilson, On maximal subgroups of Fischer's group Fi22, Math. Proc. Cambr* *. Phil. Soc. 95 (1984), 197-222 [W5] R. Wilson, The subgroup structure of the Lyons group, Math. Proc. Cambr. P* *hil. Soc. 95 (1984), 403-409 [W6] R. Wilson, The local subgroups of the Fischer groups, J. London Math. Soc.* * 36 (1987), 77-94 [W7] R. Wilson, Some subgroups of the Thompson group, J. Austral. Math. Soc. 44* * (1988), 17-32 [Y1] S. Yoshiara, The radical 2-subgroups of the sporadic simple groups J4, Co2* *, and Th, J. Algebra 233 (2000), 309-341 [Y2] S. Yoshiara, The radical 2-subgroups of some sporadic simple groups, J. Al* *gebra 248 (2002), 237-264 LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bob@math.univ-paris13.fr