CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE STUDIED BY SOLOMON AND BENSON RAN LEVI AND BOB OLIVER Abstract. A p-local finite group is an algebraic structure with a classif* *ying space which has many of the properties of p-completed classifying spaces of fin* *ite groups. In this paper, we construct a family of 2-local finite groups, which are * *exotic in the following sense: they are based on certain fusion systems over the Sylow * *2-subgroup of Spin7(q) (q an odd prime power) shown by Solomon not to occur as the 2* *-fusion in any actual finite group. Thus, the resulting classifying spaces are no* *t homotopy equivalent to the 2-completed classifying space of any finite group. As p* *redicted by Benson, these classifying spaces are also very closely related to the Dwy* *er-Wilkerson space BDI(4). As one step in the classification of finite simple groups, Ron Solomon [So] c* *onsidered the problem of classifying all finite simple groups whose Sylow 2-subgroups are* * isomor- phic to those of the Conway group Co3. The end result of his paper was that Co3* * is the only such group. In the process of proving this, he needed to consider grou* *ps G in which all involutions are conjugate, and such that the centralizer of each i* *nvolution contains a normal subgroup isomorphic to Spin7(q) with odd index, where q is an* * odd prime power. Solomon showed that such a group G does not exist. The proof of th* *is statement was also interesting, in the sense that the 2-local structure of the * *group in question appeared to be internally consistent, and it was only by analyzing its* * inter- action with the p-local structure (where p is the prime of which q is a power) * *that he found a contradiction. In a later paper [Be ], Dave Benson, inspired by Solomon's work, constructed * *certain spaces which can be thought of as the 2-completed classifying spaces which the * *groups studied by Solomon would have if they existed. He started with the spaces BDI(* *4) constructed by Dwyer and Wilkerson having the property that H*(BDI(4); F2) ~=F2[x1, x2, x3, x4]GL4(2) (the rank four Dickson algebra at the prime 2). Benson then considered, for eac* *h odd prime power q, the homotopy fixed point set of the Z-action on BDI(4) generated* * by an Ä dams operation" _q constructed by Dwyer and Wilkerson. This homotopy fixed point set is denoted here BDI4(q). In this paper, we construct a family of 2-local finite groups, in the sense o* *f [BLO2 ], which have the 2-local structure considered by Solomon, and whose classifying s* *paces are homotopy equivalent to Benson's spaces BDI4(q). The results of [BLO2 ] com* *bined with those here allow us to make much more precise the statement that these spa* *ces have many of the properties which the 2-completed classifying spaces of the gro* *ups studied by Solomon would have had if they existed. To explain what this means, * *we first recall some definitions. ___________ 1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R37, 20D0* *6, 20D20. Key words and phrases. Classifying space, p-completion, finite groups, fusion. B. Oliver is partially supported by UMR 7539 of the CNRS. 1 2 Construction of 2-local finite groups A fusion system over a finite p-group S is a category whose objects are the s* *ubgroups of S, and whose morphisms are monomorphisms of groups which include all those induced by conjugation by elements of S. A fusion system is saturated if it sa* *tisfies certain axioms formulated by Puig [Pu ], and also listed in [BLO2 , Definition* * 1.2] as well as at the beginning of Section 1 in this paper. In particular, for any finite g* *roup G and any S 2 Sylp(G), the category FS(G) whose objects are the subgroups of S and wh* *ose morphisms are those monomorphisms between subgroups induced by conjugation in G is a saturated fusion system over S. If F is a saturated fusion system over S, then a subgroup P S is called F-c* *entric if CS(P 0) = Z(P 0) for all P 0isomorphic to P in the category F. A centric linkin* *g system associated to F consists of a category L whose objects are the F-centric subgro* *ups of S, together with a functor L --! F which is the inclusion on objects, is surjec* *tive on all morphism sets and which satisfies certain additional axioms (see [BLO2 , D* *efinition 1.6]). These axioms suffice to ensure that the p-completed nerve |L|^phas all * *of the properties needed to regard it as a "classifying spaceö f the fusion system F.* * A p-local finite group consists of a triple (S, F, L), where S is a finite p-group, F is * *a saturated fusion system over S, and L is a linking system associated to F. The classifyin* *g space of a p-local finite group (S, F, L) is the p-completed nerve |L|^p(which is p-comp* *lete since |L| is always p-good [BLO2 , Proposition 1.11]). For example, if G is a finit* *e group and S 2 Sylp(G), then there is an explicitly defined centric linking system LcS* *(G) associated to FS(G), and the classifying space of the triple (S, FS(G), LcS(G))* * is the space |LcS(G)|^p' BG^p. Exotic examples of p-local finite groups for odd primes p _ i.e., examples wh* *ich do not represent actual groups _ have already been constructed in [BLO2 ], but us* *ing ad hoc methods which seemed to work only at odd primes. In this paper, we first construct a fusion system FSol(q) (for any odd prime * *power q) over a 2-Sylow subgroup S of Spin7(q), with the properties that all elements* * of order 2 in S are conjugate (i.e., the subgroups they generated are all isomorph* *ic in the category), and the "centralizer fusion system" (see the beginning of Section 1)* * of each such element is isomorphic to the fusion system of Spin7(q). We then show that * *FSol(q) is saturated, and has a unique associated linking system LcSol(q). We thus obta* *in a 2- local finite group (S, FSol(q), LcSol(q)) where by Solomon's theorem [So] (as e* *xplained in more detail in Proposition 3.4), FSol(q) is not the fusion system of any fin* *ite group. Let BSol(q) def=|LcSol(q)|^2denote the classifying space of (S, FSol(q), LcSol(* *q)). Thus, BSol(q) does not have the homotopy type of BG^2for any finite group G, but does have many of the nice properties of the 2-completed classifying space of a fini* *te group (as described in [BLO2 ]). Relating BSol(q) to BDI4(q) requires taking the ü nionö f the categories LcS* *ol(qn) for all n 1. This however is complicated by the fact that an inclusion of fi* *elds Fpm Fpn (i.e., m|n) does not induce an inclusion of cenric linking systems. H* *ence we have to replace the centric linking systems LcSol(qn) by the full subcategories* *SLccSol(qn) whose objects are those 2-subgroups which are centric in FcSol(q1 ) = n 1FcSo* *l(qn), and show that the inclusion induces a homotopy equivalence BSol0(qn) def=|LccSo* *l(qn)|^2' BSol(qn). Inclusions of fields do induce inclusions of these categories, so we * *can then S define LcSol(q1 ) def=n 1LccSol(qn), and spaces i[ j BSol(q1 ) = |LcSol(q1 )|^2' BSol0(qn) ^2. n 1 Ran Levi and Bob Oliver * * 3 The category LcSol(q1_) has an Ä dams map" _q induced by the Frobenius automor- phism x 7! xq of Fq. We then show that BSol(q1 ) ' BDI(4), the space of Dwyer a* *nd Wilkerson mentioned above; and also that BSol(q) is equivalent to the homotopy * *fixed point set of the Z-action on BSol(q1 ) generated by B_q. The space BSol(q) is t* *hus equivalent to Benson's spaces BDI4(q) for any odd prime power q. The paper is organized as follows. Two propositions used for constructing sat* *urated fusion systems, one very general and one more specialized, are proven in Sectio* *n 1. These are then applied in Section 2 to construct the fusion systems FSol(q), an* *d to prove that they are saturated. In Section 3 we prove the existence and uniquene* *ss of a centric linking systems associated to FSol(q) and study their automorphisms. * *Also in Section 3 is the proof that FSol(q) is not the fusion system of any finite g* *roup. The connections with the space BDI(4) of Dwyer and Wilkerson is shown in Section 4. Some background material on the spinor groups Spin(V, b) over fields of charact* *eristic 6= 2 is collected in an appendix. We would like to thank Dave Benson, Ron Solomon, and Carles Broto for their h* *elp while working on this paper. 1.Constructing saturated fusion systems In this section, we first prove a general result which is useful for construc* *ting satu- rated fusion systems. This is then followed by a more technical result, which i* *s designed to handle the specific construction in Section 2. We first recall some definitions from [BLO2 ]. A fusion system over a p-grou* *p S is a category F whose objects are the subgroups of F, such that Hom S(P, Q) Mor F(P, Q) Inj(P, Q) for all P, Q S, and such that each morphism in F factors as the composite of * *an F-isomorphism followed by an inclusion. We write Hom F (P, Q) = Mor F(P, Q) to emphasize that the morphisms are all homomorphisms of groups. We say that two subgroups P, Q S are F-conjugate if they are isomorphic in F. A subgroup P S is fully centralized (fully normalized) in F if |CS(P )| |CS(P 0)| (|NS(P )| * * |NS(P 0)|) for all P 0 S which is F-conjugate to P . A saturated fusion system is a fusio* *n system F over S which satisfies the following two additional conditions: (I)For each fully normalized subgroup P S, P is fully centralized and AutS (P* * ) 2 Sylp(Aut F(P )). (II)For each P S and each ' 2 Hom F(P, S) such that '(P ) is fully centralize* *d in F, if we set fi -1 N' = g 2 NS(P ) fi'cg' 2 AutS('(P )) , _ then ' extends to a homomorphism ' 2 Hom F(N', S). For example, if G is a finite group and S 2 Sylp(G), then the category FS(G) * *whose objects are the subgroups of S and whose morphisms are the homomorphisms induced by conjugation in G is a saturated fusion system over S. A subgroup P S is fu* *lly centralized in FS(G) if and only if CS(P ) 2 Sylp(CG(P )), and P is fully norma* *lized in FS(G) if and only if NS(P ) 2 Sylp(NG(P )). 4 Construction of 2-local finite groups For any fusion system F over a p-group S, and any subgroup P S, the "centra* *lizer fusion system" CF (P ) over CS(P ) is defined by setting fi 0 0 Hom CF(P)(Q, Q0) = ('|Q) fi' 2 Hom F(P Q, P Q ), '(Q) Q , '|P = IdP for all Q, Q0 CS(P ) (see [BLO2 , Definition A.3] or [Pu ] for more detail).* * We also write CF (g) = CF () for g 2 S. If F is a saturated fusion system and P is * *fully centralized in F, then CF (P ) is saturated by [BLO2 , Proposition A.6] (or [P* *u ]). Proposition 1.1. Let F be any fusion system over a p-group S. Then F is saturat* *ed if and only if there is a set X of elements of order p in S such that the follo* *wing conditions hold: (a)Each x 2 S of order p is F-conjugate to some element of X. (b)If x and y are F-conjugate and y 2 X, then there is some _ 2 Hom F(CS(x), CS* *(y)) such that _(x) = y. (c)For each x 2 X, CF (x) is a saturated fusion system over CS(x). Proof.Throughout the proof, conditions (I) and (II) always refer to the conditi* *ons in the definition of a saturated fusion system, as stated above or in [BLO2 , Def* *inition 1.2]. Assume first that F is saturated, and let X be the set of all x 2 S of order * *p such that is fully centralized. Then condition (a) holds by definition, (b) foll* *ows from condition (II), and (c) holds by [BLO2 , Proposition A.6] or [Pu ]. Assume conversely that X is chosen such that conditions (a-c) hold for F. Def* *ine fi T U = (P, x) fiP S, |x| = p, x 2 Z(P ) , some T 2 Sylp(AutF (P )) containing * *AutS(P,) where Z(P )T is the subgroup of elements of Z(P ) fixed by the action of T . Le* *t U0 U be the set of pairs (P, x) such that x 2 X. For each 1 6= P S, there is some * *x such that (P, x) 2 U (since every action of a p-group on Z(P ) has nontrivial fixed * *set); but x need not be unique. We first check that (P, x) 2 U0, P fully centralized in CF (x)=) P fully centralized in(F.1) Assume otherwise: that (P, x) 2 U0 and P is fully centralized in CF (x), but P * *is not fully centralized in F. Let P 0 S and ' 2 IsoF(P, P 0) be such that |CS(P )| <* * |CS(P 0)|. Set x0 = '(x) Z(P 0). By (b), there exists _ 2 Hom F (CS(x0), CS(x)) such th* *at _(x0) = x. Set P 00= _(P 0). Then _ O' 2 IsoCF(x)(P, P 00), and in particular* * P 00is CF (x)-conjugate to P . Also, since CS(P 0) CS(x0), _ sends CS(P 0) injectiv* *ely into CS(P 00), and |CS(P )| < |CS(P 0)| |CS(P 00)|. Since CS(P ) = CCS(x)(P ) and * *CS(P 00) = CCS(x)(P 00), this contradicts the original assumption that P is fully centrali* *zed in CF (x). By definition of U, for each (P, x) 2 U, NS(P ) CS(x) and hence AutCS(x)(P * *) = AutS(P ). By assumption, there is T 2 Sylp(Aut F(P )) such that ø(x) = x for al* *l ø 2 T ; i.e., such that T AutCF(x)(P ). In particular, it follows that 8(P, x) 2 U : Aut S(P ) 2 Sylp(Aut F(P )) () AutCS(x)(P ) 2 Sylp(Aut CF(x)(P(* *)).2) We are now ready to prove condition (I) for F; namely, to show for each P S fully normalized in F that P is fully centralized and AutS (P ) 2 Sylp(Aut F(P * *)). By definition, |NS(P )| |NS(P 0)| for all P 0F-conjugate to P . Choose x 2 Z(P )* * such that (P, x) 2 U; and let T 2 Sylp(Aut F(P )) be such that T AutS(P ) and x 2 Z(P )* *T. By (a) and (b), there is an element y 2 X and a homomorphism _ 2 Hom F(CS(x), CS(y* *)) Ran Levi and Bob Oliver * * 5 such that _(x) = y. Set P 0= _P , and set T 0= _T _-1 2 Sylp(Aut F(T 0)). Since* * T AutS(P ) by definition of U, and _(NS(P )) = NS(P 0) by the maximality assumpti* *on, 0 0 we see that T 0 AutS(P 0). Also, y 2 Z(P 0)T (T y = y since T x = x), and thi* *s shows that (P 0, y) 2 U0. The maximality of |NS(P 0)| = |NCS(y)(P 0)| implies that P * *0is fully normalized in CF (y). Hence by condition (I) for the saturated fusion system C* *F (y), together with (1) and (2), P fully centralized in F and AutS(P ) 2 Sylp(Aut F(P* * )). It remains to prove condition (II) for F. Fix 1 6= P S and ' 2 Hom F(P, S) * *such that P 0def='P is fully centralized in F, and set fi -1 0 N' = g 2 NS(P ) fi'cg' 2 AutS(P ) . _ 0 0 We must show that ' extends to some ' 2 Hom F(N', S). Choose some x 2 Z(P ) of order p which is fixed under the action of AutS(P 0), and set x = '-1(x0) 2 Z(P* * ). For all g 2 N', 'cg'-1 2 AutS(P 0) fixes x0, and hence cg(x) = x. Thus x 2 Z(N') and hence N' CS(x); and NS(P 0) CS(x0). (3) Fix y 2 X which is F-conjugate to x and x0, and choose _ 2 Hom F(CS(x), CS(y)) and _02 Hom F(CS(x0), CS(y)) such that _(x) = _0(x0) = y. Set Q = _(P ) and Q0 = _0(P 0). Since P 0is fully centralized in F, _0(P 0) = Q0, and CS(P 0) CS(x0), we have _0(CCS(x0)(P 0)) = _0(CS(P 0)) = CS(Q0) = CCS(y)(Q0). (4) Set ø = _0'_-1 2 IsoF(Q, Q0). By construction, ø(y) = y, and thus ø 2 IsoCF(y)(* *Q, Q0). Since P 0is fully centralized in F, (4) implies that Q0is fully centralized in * *CF (y). Hence condition (II), when_applied to the saturated fusion system CF (y), shows that * *ø extends to a homomorphism ø 2 Hom CF(y)(Nfi, CS(y)), where fi -1 0 Nfi= g 2 NCS(y)(Q) fiøcgø 2 AutCS(y)(Q ) . Also, for all g 2 N' CS(x) (see (3)), c_fi(_(g))= øc_(g)ø-1 = (ø_)cg(ø_)-1 = (_0')cg(_0')-1 = c_0(h)2 AutCS(y)(Q0) for some h 2 NS(P 0) such that 'cg'-1 = ch. This shows that _(N') Nfi; and al* *so (since CS(Q0) = _0(CS(P 0)) by (4)) that _ 0 0 ø(_(N')) _ (NCS(x0)(P )). We can now define _def 0 -1 _ ' = (_ ) O(ø O_)|N' 2 Hom F(N', S), _ and '|P = '. Proposition 1.1 will also be applied in a separate paper of Carles Broto & Je* *sper Møller [BM ] to give a construction of some "exotic" p-local finite groups at * *certain odd primes. Our goal now is to construct certain saturated fusion systems, by starting wi* *th the fusion system of Spin7(q) for some odd prime power q, and then adding to that t* *he automorphisms of some subgroup of Spin7(q). This is a special case of the gene* *ral problem of studying fusion systems generated by fusion subsystems, and then sho* *wing that they are saturated. We first fix some notation. If F1 and F2 are two fusio* *n systems over the same p-group S, then denotes the fusion system over S generat* *ed by F1 and F2: the smallest fusion system over S which contains both F1 and F2. More generally, if F is a fusion system over S, and F0 is a fusion system over a sub* *group 6 Construction of 2-local finite groups S0 S, then denotes the fusion system over S generated by the morphism* *s in F between subgroups of S, together with morphisms in F0 between subgroups of S0 only. In other words, a morphism in is a composite '1 '2 'k P0 ---! P1 ---! P2 ---! . . .---!Pk-1 ---! Pk, where for each i, either 'i2 Hom F(Pi-1, Pi), or 'i2 Hom F0(Pi-1, Pi) (and Pi-1* *, Pi S0). As usual, when G is a finite group and S 2 Sylp(G), then FS(G) denotes the fu* *sion system of G over S. If Aut (G) is a group of automorphisms which contains Inn(G), then FS( ) will denote the fusion system over S whose morphisms consist* * of all restrictions of automorphisms in to monomorphisms between subgroups of S. The next proposition provides some fairly specialized conditions which imply * *that the fusion system generated by the fusion system of a group G together with cer* *tain automorphisms of a subgroup of G is saturated. Proposition 1.2. Fix a finite group G, a prime p dividing |G|, and a Sylow p-su* *bgroup S 2 Sylp(G). Fix a normal subgroup Z C G of order p, an elementary abelian subg* *roup U C S of rank two containing Z such that CS(U) 2 Sylp(CG(U)), and a subgroup Aut(CG(U)) containing Inn(CG(U)) such that fl(U) = U for all fl 2 . Set S0 = CS(U) and F def=, and assume the following hold. (a)All subgroups of order p in S different from Z are G-conjugate. (b) permutes transitively the subgroups of order p in U. (c){' 2 | '(Z) = Z} = AutNG(U)(CG(U)). (d)For each E S which is elementary abelian of rank three, contains U, and is* * fully centralized in FS(G), {ff 2 AutF (CS(E)) | ff(Z) = Z} = AutG (CS(E)). (e)For all E, E0 S which are elementary abelian of rank three and contain U, * *if E and E0 are -conjugate, then they are G-conjugate. Then F is a saturated fusion system over S. Also, for any P S such that Z P* * , {' 2 Hom F(P, S) | '(Z) = Z} = Hom G(P, S). (1) Proposition 1.2 follows from the following three lemmas. Throughout the proof* *s of these lemmas, references to points (a-e) mean to those points in the hypotheses* * of the proposition, unless otherwise stated. Lemma 1.3. Under the hypotheses of Proposition 1.2, for any P S and any centr* *al subgroup Z0 Z(P ) of order p, Z 6= Z0 U =) 9 ' 2 Hom (P, S0) such that '(Z0) = Z (1) and Z0 U =) 9 _ 2 Hom G(P, S0) such that _(Z0) U. (2) Proof.Note first that Z Z(S), since it is a normal subgroup of order p in a p* *-group. Assume Z 6= Z0 U. Then U = ZZ0, and P CS(Z0) = CS(ZZ0) = CS(U) = S0 Ran Levi and Bob Oliver * * 7 since Z0 Z(P ) by assumption. By (b), there is ff 2 such that ff(Z0) = Z. S* *ince S0 2 Sylp(CG(U)), there is h 2 CG(U) such that h.ff(P ).h-1 S0; and since ch 2 AutNG(U)(CG(U)) by (c), ' def=ch Off 2 Hom (P, S0) and sends Z0 to Z. If Z0 U, then by (a), there is g 2 G such that gZ0g-1 Ur Z. Since Z is central in S, gZ0g-1 is central in gP g-1, and U is generated by Z and gZ0g-1, * *it follows that gP g-1 CG(U). Since S0 2 Sylp(CG(U)), there is h 2 CG(U) such th* *at h(gP g-1)h-1 S0; and we can take _ = chg 2 Hom G(P, S0). We are now ready to prove point (1) in Proposition 1.2. Lemma 1.4. Assume the hypotheses of Proposition 1.2, and let F = be the fusion system generated by G and . Then for all P, P 0 S which contain* * Z, {' 2 Hom F(P, P 0) | '(Z) = Z} = Hom G(P, P 0). Proof.Upon replacing P 0by '(P ) P 0, we can assume that ' is an isomorphism,* * and thus that it factors as a composite of isomorphisms '1 '2 '3 'k-1 'k 0 P = P0 ---!~ P1 ---! P2 ---! . . .---! Pk-1 ---! Pk = P , = ~= ~= ~= ~= where for each i, 'i2 Hom G(Pi-1, Pi) or 'i2 Hom (Pi-1, Pi). Let Zi Z(Pi) be * *the subgroups of order p such that Z0 = Zk = Z and Zi= 'i(Zi-1). To simplify the discussion, we say that a morphism in F is of type (G) if it * *is given by conjugation by an element of G, and of type ( ) if it is the restricti* *on of an automorphism in . More generally, we say that a morphism is of type (G, )* * if it is the composite of a morphism of type (G) followed by one of type ( ), etc.* * We regard IdP, for all P S, to be of both types, even if P S0. By definition, * *if any nonidentity isomorphism is of type ( ), then its source and image are both cont* *ained in S0 = CS(U). For each i, using Lemma 1.3, choose some _i2 Hom F(PiU, S) such that _i(Zi) =* * Z. More precisely, using points (1) and (2) in Lemma 1.3, we can choose _i to be o* *f type ( ) if Zi U (the inclusion if Zi = Z), and to be of type (G, ) if Z U. Set Pi0= _i(Pi). To keep track of the effect of morphisms on the subgroups Zi, we w* *rite them as morphisms between pairs, as shown below. Thus, ' factors as a composite* * of isomorphisms _-1i-1 'i _i 0 (Pi0-1, Z) -----! (Pi-1, Zi-1) -----! (Pi, Zi) -----! (Pi, Z). If 'iis of type (G), then this composite (after replacing adjacent morphisms of* * the same type by their composite) is of type ( , G, ). If 'i is of type ( ), then the c* *omposite is again of type ( , G, ) if either Zi-1 U or Zi U, and is of type ( , G, ,* * G, ) if neither Zi-1nor Zi is contained in U. So we are reduced to assuming that ' is o* *f one of these two forms. Case 1: Assume first that ' is of type ( , G, ); i.e., a composite of isomorph* *isms of the form '1 '2 '3 (P0, Z) ----! (P1, Z1) ----! (P2, Z2) ----! (P3, Z). ( ) (G) ( ) Then Z1 = Z if and only if Z2 = Z because '2 is of type (G). If Z1 = Z2 = Z, th* *en '1 and '3 are of type (G) by (c), and the result follows. 8 Construction of 2-local finite groups If Z1 6= Z 6= Z2, then U = ZZ1 = ZZ2, and thus '2(U) = U. Neither '1 nor '3 c* *an be the identity, so Pi S0 = CS(U) for all i by definition of Hom (-, -), and* * hence '2 is of type ( ) by (c). It follows that ' 2 Iso (P0, P3) sends Z to itself, * *and is of type (G) by (c) again. Case 2: Assume now that ' is of type ( , G, , G, ); more precisely, that it * *is a composite of the form '1 '2 '3 '4 '5 (P0, Z) ---! (P1, Z1) ---! (P2, Z2) ---! (P3, Z3) ---! (P4, Z4) ---! (P5,* * Z), ( ) (G) ( ) (G) ( ) where Z2, Z3 U. Then Z1, Z4 U and are distinct from Z, and the groups P0, P1, P4, P5 all contain U since '1 and '5 (being of type ( )) leave U invari* *ant. In particular, P2 and P3 contain Z, since P1 and P4 do and '2, '4 are of type (* *G). We can also assume that U P2, P3, since otherwise P2\U = Z or P3\U = Z, '3(Z) = * *Z, and hence '3 is of type (G) by (c) again. Finally, we assume that P2, P3 S0 =* * CS(U), since otherwise '3 = Id. Let Ei Pibe the rank three elementary abelian subgroups defined by the requi* *re- ments that E2 = UZ2, E3 = UZ3, and 'i(Ei-1) = Ei. In particular, Ei Z(Pi) for i = 2, 3 (since Zi Z(Pi), and U Z(Pi) by the above remarks); and hence Ei Z(Pi) for all i. Also, U = ZZ4 '4(E3) = E4 since '4(Z) = Z, and thus U = '5(U) E5. Via similar considerations for E0 and E1, we see that U Eifor* * all i. Set H = CG(U) for short. Let E3 be the set of all elementary abelian subgroup* *s E S of rank three which contain U, and with the property that CS(E) 2 Sylp(CH (E)* *). Since CS(E) CS(U) = S0 2 Sylp(H), the last condition implies that E is fully centralized in the fusion system FS0(H). If E S is any rank three elementary * *abelian subgroup which contains U, then there is some a 2 H such that E0 = aEa-1 2 E3, since FS0(H) is saturated and U C H. Then ca 2 IsoG(E, E0) \ Iso(E, E0) by (c).* * So upon composing with such isomorphisms, we can assume that Ei 2 E3 for all i, and also that 'i(CS(Ei-1)) = CS(Ei) for each i. _ In this way, ' can be assumed to extend to an F-isomorphism ' from CS(E0) to CS(E5) which sends Z to itself. By (e), the rank three subgroups Ei are all* * G- conjugate to each other. Choose g 2 G such that gE5g-1 = E0. Then g.CS(E5).g-1 and CS(E0) are both Sylow p-subgroups of_CG(E0), so there is h 2 CG(E0) such th* *at (hg)CS(E5)(hg)-1 = CS(E0). By (d), chgO '2 AutF (CS(E0)) is of type (G); and th* *us ' 2 IsoG(P0, P5). To finish the proof of Proposition 1.2, it remains only to show: Lemma 1.5. Under the hypotheses of Proposition 1.2, the fusion system F generat* *ed by FS(G) and FS0( ) is saturated. Proof.We apply Proposition 1.1, by letting X be the set of generators of Z. Con* *dition (a) of the proposition (every x 2 S of order p is F-conjugate to an element of * *X) holds by Lemma 1.3. Condition (c) holds since CF (Z) is the fusion system of the gro* *up CG(Z) by Lemma 1.4, and hence is saturated by [BLO2 , Proposition 1.3]. It remains to prove condition (b) of Proposition 1.1. We must show that if y,* * z 2 S are F-conjugate and = Z, then there is _ 2 Hom F(CS(y), CS(z)) such that _(* *y) = z. If y =2U, then by Lemma 1.3(2), there is ' 2 Hom F(CS(y), S0) such that '(y)* * 2 U. If y 2 Ur Z, then by Lemma 1.3(1), there is ' 2 Hom F(CS(y), S0) such that '(y)* * 2 Z. We are thus reduced to the case where y, z 2 Z (and are F-conjugate). Ran Levi and Bob Oliver * * 9 In this case, then by Lemma 1.4, there is g 2 G such that z = gyg-1. Since Z * *C G, [G:CG(Z)] is prime to p, so S and gSg-1 are both Sylow p-subgroups of CG(Z), and hence are CG(Z)-conjugate. We can thus choose g such that z = gyg-1 and gSg-1 =* * S. Since CS(y) = CS(z) = S (Z Z(S) since it is a normal subgroup of order p), th* *is shows that cg 2 IsoG(CS(y), CS(z)), and finishes the proof of (b) in Propositio* *n 1.1. 2. A fusion system of a type considered by Solomon The main result of this section and the next is the following theorem: Theorem 2.1. Let q be an odd prime power, and fix S 2 Syl2(Spin7(q)). Let z 2 Z(Spin7(q)) be the central element of order 2. Then there is a saturated fusion* * system F = FSol(q) which satisfies the following conditions: (a)CF (z) = FS(Spin7(q)) as fusion systems over S. (b)All involutions of S are F-conjugate. Furthermore, there is a unique centric linking system L = LcSol(q) associated t* *o F. Theorem 2.1 will be proven in Propositions 2.11 and 3.3. Later, at the end of* * Section 3, we explain why Solomon's theorem [So] implies that these fusion systems are * *not the fusion systems of any finite groups, and hence that the spaces BSol(q) are * *not homotopy equivalent to the 2-completed classifying spaces of any finite groups. Background results needed for computations in Spin(V, b) have been collected * *in Appendix A. We focus attention here on SO7(q) and Spin7(q). In fact, since we w* *ant to compare the constructions over Fq with those over_its field extensions,_most* * of the constructions will first be made in the groups SO7(F q) and Spin7(F q). We now fix, for the rest of the_section, an odd prime power q. It will be con* *venient to write Spin7(q1 ) def=Spin7(F q), etc. In order to make certain computations* * more explicit, we set _ _ _ V1 = M2(F q) M02(F q) ~=(F q)7 and b(A, B) = det(A) + det(B) (where M02(-) is the group of (2 x 2) matrices of trace zero), and for each n * * 1 set Vn = M2(Fqn) M02(Fqn) V1 . Then b is a nonsingular quadratic form on V1 and on Vn. Identify SO7(q1 ) = SO(V1 , b) and SO7(qn)_= SO(Vn, b), and similarly_f* *or Spin7(qn) Spin7(q1 ). For all ff 2 Spin(M2(F q), det) and fi 2 Spin(M02(F q),* * det), we write ff fi for their image in Spin7(q1 ) under the natural homomorphism '4,3:Spin4(q1 ) x Spin3(q1 ) -----! Spin7(q1 ). There are isomorphisms ~= 1 1 ~= 1 eæ4:SL2(q1 ) x SL2(q1 ) --! Spin4(q ) and eæ3:SL2(q ) --! Spin3(q ) which are defined explicitly in Proposition A.5, and which restrict to isomorph* *isms SL2(qn) x SL2(qn) ~=Spin4(qn) and SL2(qn) ~=Spin3(qn) for each n. Let z = eæ4(-I, -I) 1 = 1 eæ3(-I) 2 Z(Spin7(q)) 10 Construction of 2-local finite groups denote the central element of order two, and set z1 = eæ4(-I, I) 1 2 Spin7(q). Here, 1 2 Spink(q) (k = 3, 4) denotes the identity element. Define U = . Definition 2.2. Define ! :SL2(q1 )3 -----! Spin7(q1 ) by setting !(A1, A2, A3) = eæ4(A1, A2) eæ3(A3) for A1, A2, A3 2 SL2(q1 ). Set H(q1 ) = !(SL2(q1 )3) and [[A1, A2, A3]] = !(A1, A2, A3) . Since eæ3and eæ4are isomorphisms, Ker(!) = Ker('4,3), and thus Ker(!) = <(-I, -I, -I)>. In particular, H(q1 ) ~=(SL2(q1 )3)={ (I, I, I)}. Also, z = [[I, I, -I]] and z1 = [[-I, I, I]], and thus U = [[ I, I, I]] (with all combinations of signs). For each 1 n < 1, the natural homomorphism Spin7(qn) ------! SO7(qn) has kernel and cokernel both of order 2. The image of this homomorphism is the commutator subgroup 7(qn) C SO7(qn),_which is partly described by Lemma A.4(a). In contrast, since all elements of F qare squares, the natural homomorphism from Spin7(q1 ) to SO7(q1 ) is surjective. Lemma 2.3. There is an element ø 2 NSpin7(q)(U) of order 2 such that ø.[[A1, A2, A3]].ø-1 = [[A2, A1, A3]] (1) for all A1, A2, A3 2 SL2(q1 ). _ Proof.Let ø 2 SO7(q) be the involution defined by setting _ ø(X, Y ) = (-`(X), -Y ) _ _ for (X, Y ) 2 V1 = M2(F q) M02(F q), where ` abcd= -d-bca. _ _ Let ø 2 Spin7(q1 ) be a lifting of ø. The (-1)-eigenspace of ø on V1 has ortho* *gonal basis (I, 0) , 0, 100-1 , 0, 0110 , 0, 0-110 , and in particular_has discriminant 1 with respect to this basis. Hence by Lemm* *a _ A.4(a), ø 2 7(q), and so ø 2 Spin7(q). Since in addition, the (-1)-eigenspace * *of ø is 4-dimensional, Lemma A.4(b) applies to show that ø2 = 1. By definition of the isomorphisms eæ3and eæ4, for all Ai 2 SL2(q1 ) (i = 1, 2* *, 3) and all (X, Y ) 2 V1 , [[A1, A2, A3]](X, Y ) = (A1XA-12, A3Y A-13). Ran Levi and Bob Oliver * * 11 * * _ Here, Spin7(q1 ) acts on V1 via its projection to SO7(q1 ). Also, for all X, Y * *2 M2(F q), t -1 `(X) = -0110.X . 0-110 and in particular `(XY ) = `(Y ).`(X); and `(X) = X-1 if det(X) = 1. Hence for all A1, A2, A3 2 SL2(q1 ) and all (X, Y* * ) 2 V1 , -1 -1 -1 ø.[[A1, A2, A3]].ø (X,=Yø)(-A1.`(X).A2 , -A3Y A3 ) = (A2XA-11, A3Y A-13) = [[A2, A1, A3]](X, Y ). This shows that (1) holds modulo = Z(Spin7(q1 )). We thus have two automor- phisms of H(q1 ) ~= (SL2(q1 )3)={ (I, I, I)} _ conjugation by ø and the permuta- tion automorphism _ which are liftings of the same automorphism of H(q1 )=. Since H(q1 ) is perfect, each automorphism of H(q1 )= has at most one lifting to an automorphism of H(q1 ), and thus (1) holds. Also, since U is the subgroup of all elements [[ I, I, I]] with all combinations of signs, formula (1) show* *s that ø 2 NSpin7(q)(U). Definition 2.4. For each n 1, set H(qn) = H(q1 ) \ Spin7(qn) and H0(qn) = !(SL2(qn)3) H(qn). Define n = Inn(H(qn)) o b 3 Aut(H(qn)), where b 3denotes the group of permutation automorphisms fi b 3= [[A1, A2, A3]] 7! [[Aff1, Aff2, Aff3]] fioe 2 3 Aut(H(qn)) . n 1 For each n, letn_q be the automorphism of Spin7(q ) induced by thenfield is* *omor- phism (q 7! qp ). By Lemma A.3, Spin7(qn) is the fixed subgroup of _q . Hence e* *ach element of H(qn) is of the form [[A1,nA2, A3]], where either Ai2 SL2(qn) for ea* *ch i (and the element lies in H0(qn)), or _q (Ai) = -Ai for each i. This shows that H0(qn* *) has index 2 in H(qn). The goal is now to choose öc mpatible" Sylow subgroups S(qn) 2 Syl2(Spin7(qn)) (all n 1) contained in N(H(qn)), and let FSol(qn) be the fusion system over S* *(qn) generated by conjugation in Spin7(qn) and by restrictions of n. Proposition 2.5. The following hold for each n 1. (a)H(qn) = CSpin7(qn)(U). (b)NSpin7(qn)(U) = NSpin7(qn)(H(qn)) = H(qn).<ø>, and contains a Sylow 2-subgro* *up of Spin7(qn). _ _ Proof.Let_z1 2 SO7(q) be the image of z1 2 Spin7(q). Set V- = M2(F q) and V+ = _ M02(F q): the eigenspaces of z1acting on V . By Lemma A.4(c), CSpin7(q1()U) = CSpin7(q1()z1) _ 1 is the group of all elements ff 2 Spin7(q1 ) whose image ff2 SO7(q ) has the f* *orm _ ff= ff- ff+ where ff 2 SO(V ). In other words, 1 1 1 3 1 CSpin7(q1()U) = '4,3Spin4(q ) x Spin3(q ) = !(SL2(q ) ) = H(q ). Furthermore, since øz1ø-1 = ø[[-I, I, I]]ø-1 = [[I, -I, I]] = zz1 12 Construction of 2-local finite groups by Lemma 2.3, and since any element of NSpin7(q1()U) centralizes z, conjugation* * by ø generates OutSpin7(q1()U). Hence NSpin7(q1()U) = H(q1 ).<ø>. Point (a), and the first part of point (b), now follow upon taking intersection* *s with Spin7(qn). If NSpin7(qn)(U) did not contain a Sylow 2-subgroup of Spin7(qn), then since * *every noncentral involution of Spin7(qn) is conjugate to z1 (Proposition A.8), the Sy* *low 2- subgroups of Spin7(q) would have no normal subgroup isomorphic to C22. By a the* *orem of Hall (cf. [Go , Theorem 5.4.10]), this would imply that they are cyclic, di* *hedral, quaternion, or semidihedral. This is clearly not the case, so NSpin7(qn)(U) mus* *t contain a Sylow 2-subgroup of Spin7(q), and this finishes the proof of point (b). Alternatively, point (b) follows from the standard formulas for the orders of* * these groups (cf. [Ta , pp.19,140]), which show that |Spin7(qn)|_q9n(q6n - 1)(q4n - 1)(q2n - 1)6n 4n 2n i q2n + 1j = ___________________________= q (q + q + 1) _______ |H(qn).<ø>| 2.[qn(q2n - 1)]3 2 is odd. We next fix, for each n, a Sylow 2-subgroup of Spin7(qn) which is contained in H(qn).<ø> = NSpin7(qn)(U). Definition 2.6. Fix elements A, B 2 SL2(q) such that ~= Q8 (a quaternion group of order 8), and set bA= [[A, A, A]] and bB= [[B, B, B]]. Let C(q1 ) CS* *L2(q1()A) be the subgroup of elements of 2-power order in the centralizer (which is abeli* *an), and set Q(q1 ) = . Define S0(q1 ) = !(Q(q1 )3) H0(q1 ) and S(q1 ) = S0(q1 ).<ø> H(q1 ) Spin7(q1* * ). Here, ø 2 Spin7(q) is the element of Lemma 2.3. Finally, for each n 1, define C(qn) = C(q1 ) \ SL2(qn), Q(qn) = Q(q1 ) \ SL2(qn), S0(qn) = S0(q1 ) \ Spin7(qn), and S(qn) = S(q1 ) \ Spin7(qn). Since the two eigenvalues of A are distinct, its centralizer in SL2(q1 ) is c* *onjugate to the subgroup of diagonal matrices, which is abelian. Thus C(q1 ) is conjugate t* *o the subgroup of diagonal matrices of 2-power order. This shows that each finite sub* *group of C(q1 ) is cyclic, and that each finite subgroup of Q(q1 ) is cyclic or quate* *rnion. Lemma 2.7. For all n, S(qn) 2 Syl2(Spin7(qn)). Proof.By [Sz, 6.23], A is contained in a cyclic subgroup of order qn - 1 or qn * *+ 1 (depending on which of them is divisible by 4). Also, the normalizer of this c* *yclic subgroup is a quaternion group of order 2(qn 1), and the formula |SL2(qn)| = qn* *(q2n- 1) shows that this quaternion group has odd index. Thus by construction, Q(qn) * *is a Sylow 2-subgroup of SL2(qn). Hence !(Q(qn)3) is a Sylow 2-subgroup of H0(qn), * *so !(Q(q1 )3) \ Spin7(qn) is a Sylow 2-subgroup of H(qn). It follows that S(qn) is* * a Sylow 2-subgroup of H(qn).<ø>, and hence also of Spin7(qn) by Proposition 2.5(b). Following the notation of Definition A.7, we say that an elementary abelian 2- subgroup E Spin7(qn) has type I if its eigenspaces all have square discrimina* *nt, and has type II otherwise. Let Er be the set of elementary abelian subgroups of* * rank r in Spin7(qn) which contain z, and let EIrand EIIrbe the sets of those of type* * I or Ran Levi and Bob Oliver * * 13 II, respectively. In Proposition A.8, we show that there are two conjugacy clas* *ses of subgroups in EI4and one conjugacy class of subgroups in EII4. In Proposition A.* *9, an invariant xC(E) 2 E is defined, for all E 2 E4 (and where C is one of the conju* *gacy classes in EI4) as a tool for determining the conjugacy class of a subgroup. Mo* *re pre- cisely, E has type I if and only if xC(E) 2 , and E 2 C if and only if xC(E)* * = 1. The next lemma provides some more detailed information about the rank four subgroups and these invariants. Recall that we define bA= [[A, A, A]] and bB= [[B, B, B]]. Lemma 2.8. Fix n 1, set E* = S(qn), and let C be the Spin7(q* *n)- conjugacy class of E*. Let EU4be the set of all elementary abelian subgroups E * * S(qn) of rank 4 which contain U = . Fix a generator X 2 C(qn) (the 2-power tor* *sion in CSL2(qn)(A)), and choose Y 2 C(q2n) such that Y 2= X. Then the following hol* *d. (a)E* has type I. (b)EU4= Eijk, E0ijk| i, j, k 2 Z (a finite set), where Eijk= and E0ijk= . (c)xC(Eijk) = [[(-I)i, (-I)j, (-I)k]] and xC(E0ijk) = [[(-I)i, (-I)j, (-I)k]].b* *A. (d)All of the subgroups E0ijkhave type II. The subgroup Eijkhas type I if and o* *nly if i j (mod 2), and lies in C (is conjugate to E*) if and only if i j k (* *mod 2). The subgroups E000, E001, and E100thus represent the three conjugacy cla* *sses of rank four elementary abelian subgroups of Spin7(qn) (and E* = E000). (e)For any ' 2 n Aut(H(qn)) (see Definition 2.4), if E0, E002 EU4are such th* *at '(E0) = E00, then '(xC(E0)) = xC(E00). Proof.(a) The set (I, 0) , (A, 0) , (B, 0) , (AB, 0) , (0, A) , (0, B) , (0, AB) is a basis of eigenvectors for the action of E* on Vn = M2(Fqn) M02(Fqn). (Si* *nce the matrices A, B, and AB all have order 4 and determinant one, each has as eigenva* *lues the two distinct fourth roots of unity, and hence they all have trace zero.) Si* *nce all of these have determinant one, E* has type I by definition. (b) Consider the subgroups i j k i j k fi R0 = !(C(q1 )3) \ S(qn) = [[X , X , X ]], [[X Y, X Y, X Y ]] fii, j, k 2* * Z and R1 = CS(qn)() = R0.. Clearly, each subgroup E 2 EU4is contained in CS(qn)(U) = S0(qn) = R0.<[[Bi, Bj, Bk]]>. All involutions in this subgroup are contained in R1 = R0.<[[B, B, B]]>, and th* *us E R1. Hence E \ R0 has rank 3, which implies that E (the 2-torsion i* *n R0). Since all elements of order two in the coset R0.bBhave the form [[XiB, XjB, XkB]] or [[XiY B, XjY B, XkY B]] for some i, j, k, this shows that E must be one of the groups Eijkor E0ijk. (N* *ote in particular that E* = E000.) 14 Construction of 2-local finite groups (c) By Proposition A.9(a),nthe element xC(E) 2 E is characterized uniquely by * *the property that xC(E) = g-1_q (g) for some g 2 Spin7(q1 ) such that gEg-1 2 C. We now apply this explicitly to the subgroups Eijkand E0ijk. For each i, Y -i(XiB)Y i= Y -2iXiB = B. Hence for each i, j, k, [[Y i, Y j, Y k]]-1.Eijk.[[Y i, Y j, Y k]] = E* and n _q ([[Y i, Y j, Y k]]) = [[Y i, Y j, Y k]].[[(-I)i, (-I)j, (-I)k]* *]. Hence xC(Eijk) = [[(-I)i, (-I)j, (-I)k]]. Similarly, if we choose Z 2 CSL2(q1()A) such that Z2 = Y , then for each i, (Y iZ)-1(XiY B)(Y iZ) = B. Hence for each i, j, k, [[Y iZ, Y jZ, Y kZ]]-1.E0ijk.[[Y iZ, Y jZ, Y kZ]] = E*. n Since _q (Z) = ZA, n i j k i j k i j k _q ([[Y Z, Y Z, Y Z]]) = [[Y Z, Y Z, Y Z]].[[(-I) A, (-I) A, (-I) A]], and hence xC(E0ijk) = [[(-I)iA, (-I)jA, (-I)kA]]. (d) This now follows immediately from point (c) and Proposition A.9(b,c). (e) By Definition 2.4, n is generated by Inn(H(qn)) and the permutations of t* *he three factors in H(q1 ) ~= (SL2(q1 )3)={ (I, I, I)}. If ' 2 n is a permutatio* *n au- tomorphism, then it permutes the elements of EU4, and preserves the elements xC* *(-) by the formulas in (c). If ' 2 Inn(H(qn)) and '(E0) = E00for E0, E002 EU4, then '(xC(E0)) = xC(E00) by definition of xC(-); and so the same property holds for * *all elements of n. Following the notation introduced in Section 1, for P, Q S(qn), Hom Spin7(q* *n)(P, Q) denotes the set of homomorphisms from P to Q induced by conjugation by some element of Spin7(qn). Also, if P, Q S(qn) \ H(qn), Hom n(P, Q) denotes the * *set of homomorphisms induced by restriction of an element of n. Let Fn = FSol(qn) * *be the fusion system over S(qn) generated by Spin7(qn) and n. In other words, for* * each P, Q S(qn), Hom Fn(P, Q) is the set of all composites '1 '2 'k P = P0 ---! P1 ---! P2 ---! . . .---!Pk-1 ---! Pk = Q, where Pi S(qn) for all i, and each 'i lies in Hom Spin7(qn)(Pi-1, Pi) or (if * *Pi-1, Pi H(qn)) Hom n(Pi-1, Pi). This clearly defines a fusion system over S(qn). Proposition 2.9. Fix n 1. Let E S(qn) be an elementary abelian subgroup of rank 3 which contains U, and such that CS(qn)(E) 2 Syl2(CSpin7(qn)(E)). Then {' 2 AutFn(CS(qn)(E)) | '(z) = z} = AutSpin7(qn)(CS(qn)(E)). (1) Proof.Set Spin = Spin7(qn), S = S(qn), = n, and F = Fn for short. Consider the subgroups R0 = R0(qn) def=!(C(q1 )3) \ S and R1 = R1(qn) def=CS() = . Ran Levi and Bob Oliver * * 15 Here, R0 is generated by elements ofnthe form [[X1, X2, X3]], where either Xi2 * *C(qn), or X1 = X2 = X3 = X 2 C(q2n) and _q (X) = -X. Also, C(qn) 2 Syl2(CSL2(qn)(A)) is cyclic of order 2k 4, where 2k is the largest power which divides qn 1; * *and C(q2n) is cyclic of order 2k+1. So R0 ~=(C2k)3 and R1 = R0 o , where bB= [[B, B, B]] has order 2 and acts on R0 via (g 7! g-1). Note that = <[[ I, I, I]], [[A, A, A]]> ~=C32 is the 2-torsion subgroup of R0. We claim that R0 is the only subgroup of S isomorphic to (C2k)3. (2) To see this, let R0 S be any subgroup isomorphic to (C2k)3, and let E0 ~=C32be* * its 2-torsion subgroup. Recall that for any 2-group P , the Frattini subgroup Fr(P * *) is the subgroup generated by commutators and squares in P . Thus E0 Fr(R0) Fr(S) (note that [[B, B, I]] = (ø.[[B, I, I]])2). Any elementary abelian subgroup of * *rank 4 in Fr(S) would have to contain (the 2-torsion in R0 ~=C32k), and this is im* *possi- ble since no element of the coset R0.[[B, B, I]] commutes with Ab. Thus, rk(Fr* *(S)) = 3. Hence U E0, since otherwise would be an elementary abelian sub- group of Fr(S) of rank 4. This in turn implies that R0 CS(U), and hence that E0 Fr(CS(U)) R0. Thus E0 = (the 2-torsion in R0 again). Hence R0 CS() = , and it follows that R0= R0. This finishes the proof* * of (2). Choose generators x1, x2, x3 2 R0 as follows. Fix X 2 CSL2(q1()A) of order 2k* *, and Y 2 CSL2(q2n)(A) of order 2k+1 such that Y 2= X. Set x1 = [[I, I, X]], x2 = [[X* *, I, I]], k-1 2k-1 2k-1 and x3 = [[Y, Y, Y ]]. Thus, x21 = z, x2 = z1, and (x3) = bA. Now let E S(qn) be an elementary abelian subgroup of rank 3 which contains * *U, and such that CS(qn)(E) 2 Syl2(CSpin(E)). In particular, E R1 = CS(qn)(U). Th* *ere are two cases to consider: that where E R0 and that where E R0. Case 1: Assume E R0. Since R0 is abelian of rank 3, we must have E = , the 2-torsion subgroup of R0, and CS(E) = R1. Also, by (2), neither R0 nor R1 * *is isomorphic to any other subgroup of S; and hence ff AutF (Ri) = AutSpin(Ri), Aut (Ri) for i = 0, 1. (4) By Proposition A.8, AutSpin(E) is the group of all automorphisms of E which s* *end z to itself. In particular, since H(qn) = CSpin(U), AutH(qn)(E) is the group of* * all auto- morphisms of E which are the identity on U. Also, = Inn(H(qn)).b 3, where b 3* *sends bA= [[A, A, A]] to itself and permutes the nontrivial elements of U = {[[ I, I* *, I]]}. Hence Aut (E) is the group of all automorphisms which send U to itself. So if* * we identify Aut(E) ~=GL3(Z=2) via the basis {z, z1, bA}, then Aut Spin(E) = T1 def=GL12(Z=2) = (aij) 2 GL3(Z=2) | a21= a31= 0 and Aut (E) = T2 def=GL21(Z=2) = (aij) 2 GL3(Z=2) | a31= a32= 0 . 16 Construction of 2-local finite groups By (2) (and since E is the 2-torsion in R0), NSpin(E) = NSpin(R0) and {fl 2 | fl(E) = E} = {fl 2 | fl(R0) = R0* *}. Since CSpin(E) = CSpin(R0)., the only nonidentity element of Aut Spin(R0) o* *r of Aut (R0) which is the identity on E is conjugation by bB, which is -I. Hence re* *striction from R0 to E induces isomorphisms Aut Spin(R0)={ I} ~=AutSpin(E) and Aut (R0)={ I} ~=Aut (E). Upon identifying Aut(R0) ~=GL3(Z=2k) via the basis {x1, x2, x3}, these can be r* *egarded as sections ~i:Ti-----! GL3(Z=2k)={ I} = SL3(Z=2k) x {~I | ~ 2 (Z=2k)*}={ I} of the natural projection from GL3(Z=2k)={ I} to GL3(Z=2), which agree on the g* *roup T0 = T1 \ T2 of upper triangular matrices. We claim that ~1 and ~2 both map trivially to the second factor. Since this f* *actor is abelian, it suffices to show that T0 is generated by [T1, T1] \ T0 and [T2, * *T2] \ T0, and that each Ti is generated by [Ti, Ti] and T0 _ and this is easily checked. (Not* *e that T1 ~=T2 ~= 4.) By carrying out the above procedure over the field Fq2n, we see that both of * *these sections ~i can be lifted further to SL3(Z=2k+1) (still agreeing on T0). So by * *Lemma A.10, there is a section ~: GL3(Z=2) -----! SL3(Z=2k) which extends both ~1 and ~2. By (4), AutF (R0) = Im(~).<- I>. We next identify Aut F(R1). By Lemma 2.8(a), E* def= Spin7(q* *n) is a subgroup of rank 4 and type I. So by Proposition A.8, Aut Spin(E*) contain* *s all automorphisms of E* ~= C42which send z 2 Z(Spin) to itself. Hence for any x 2 NSpin(R1), since cx(z) = z, there is x1 2 NSpin(E*) such that cx1|E = cx|E (i.e* *., xx-112 CSpin(E)) and cx1(Bb) = bB(i.e., [x1, bB] = 1). Set x2 = xx-11. Since CSpin(U) = H(qn) Im(!), we see that CSpin(E) = K0., where K0 = !(CSL2(q1()A)3) \ Spin is abelian, R0 2 Syl2(K0), and bBacts on K0 by inversion. Upon replacing x1 by * *bBx1 and x2 by x2bB-1if necessary, we can assume that x2 2 K0. Then [x2, bB] = x2.(Bbx2bB-1)-1 = x22, while by the original choice of x, x1 we have [x2, bB] = [xx-11, bB] = [x, bB] 2 R0. Thus x222 R0 2 Syl2(K0), and hence x2 2 R0 R1. Since x = x2x1 was an arbitrary element of NSpin(R1), this shows that NSpin(R1) R1.CSpin(Bb), and hence that Aut Spin(R1) = Inn(R1).{' 2 AutSpin(R1) | '(Bb) = bB}. (5) Since Aut (R1) is generated by its intersection with AutSpin(R1) and the grou* *p b 3 which permutes the three factors in H(q1 ) (and since the elements of b 3all fi* *x bB), we also have Aut (R1) = Inn(R1).{' 2 Aut (R1) | '(Bb) = bB}. Ran Levi and Bob Oliver * * 17 Together with (4) and (5), this shows that AutF (R1) is generated by Inn(R1) to* *gether with certain automorphisms of R1 = R0.which send bBto itself. In other word* *s, fi Aut F(R1) = Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 AutF (R0) fi = Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 ~(GL3(Z=2)) . Thus fi ' 2 AutF (R1) fi'(z) = z fi = Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 ~(T1) = AutSpin(R0) = AutSpin(R1), the last equality by (5); and (1) now follows. Case 2: Now assume that E R0. By assumption, U E (hence E CS(E) CS(U)), and CS(E) is a Sylow subgroup of CSpin(E). Since CS(E) is not isomorphic to R1 = CS() (by (2)), this shows that E is not Spin-conjugate to . By Proposition A.8, Spin contains exactly two conjugacy classes of rank 3 subgr* *oups containing z, and thus E must have type II. Hence by Proposition A.8(d), CS(E) * *is elementary abelian of rank 4, and also has type II. Let C be the Spin7(qn)-conjugacy class of the subgroup E* = ~=C42,* * which by Lemma 2.8(a) has type I. Let E0 be the set of all subgroups of S which are e* *le- mentary abelian of rank 4, contain U, and are not in C. By Lemma 2.8(e), for a* *ny ' 2 Iso (E0, E00) and any E0 2 E0, E00def='(E0) 2 E0, and ' sends xC(E0) to xC(* *E00). The same holds for ' 2 IsoSpin(E0, E00) by definition of the elements xC(-) (Pr* *oposi- tion A.9). Since CS(E) 2 E0, this shows that all elements of Aut F(CS(E)) send* * the element xC(CS(E)) to itself. By Proposition A.9(c), Aut Spin(CS(E)) is the gro* *up of automorphisms which are the identity on the rank two subgroup ; a* *nd (1) now follows. One more technical result is needed. Lemma 2.10. Fix n 1, and let E, E0 S(qn) be two elementary abelian subgroups* * of rank three which contain U, and which are n-conjugate. Then E and E0are Spin7(* *qn)- conjugate. Proof.By [Sz, 3.6.3(ii)], -I is the only element of order 2 in SL2(q1 ). Consid* *er the sets fi J1 = X 2 SL2(qn) fiX2 = -I and fi n J2 = X 2 SL2(q2n) fi_q (X) = -X, X2 = -I . n qn Here, as usual, _q is induced by the field automorphism (x 7! x ). All elemen* *ts in J1 are SL2(q)-conjugate (this follows, for example, from [Sz, 3.6.23]), and we * *claim the same is true for elements of J2. n Let SL*2(qn) be the group of all elements X 2 SL2(q2n) such that _q (X) = X. This is a group which contains SL2(qn) with index 2. Let k be such that the Sy* *low 2-subgroups of SL2(qn) have order 2k; then k 3 since |SL2(qn)| = qn(q2n - 1).* * Any S 2 Syl2(SL*2(qn)) is quaternion of order 2k+1 16 (see [Go , Theorem 2.8.3]) * *and its intersection with SL2(qn) is quaternion of order 2k, so all elements in S \* * J2 are S-conjugate. It follows that all elements of J2 are SL*2(qn)-conjugate. If X, X* *0 2 J2 and X0 = gXg-1 for g 2 SL*2(qn), then either g 2 SL2(qn) or gX 2 SL2(qn), and in either case X and X0 are conjugate by an element of SL2(qn). 18 Construction of 2-local finite groups By Proposition 2.5(a), E, E0 CSpin7(qn)(U) = H(qn) def=!(SL2(q1 )3) \ Spin7(qn). Thus E = and E0 = , where the* * Xi are all in J1 or all in J2, and similarly for the X0i. Also, since E and E0 are n-con* *jugate (and each element of n leaves U = invariant), the Xi and X0imust all b* *e in the same set J1 or J2. Hence they are all SL2(qn)-conjugate, and so E and E0 a* *re Spin7(qn)-conjugate. We are now ready to show that the fusion systems Fn are saturated, and satisf* *y the conditions listed in Theorem 2.1. Proposition 2.11. For a fixed odd prime power q, let S(qn) S(q1 ) Spin7(q1 ) be as defined above. Let z 2 Z(Spin7(q1 )) be the central element of order 2. * * Then for each n, Fn = FSol(qn) is saturated as a fusion system over S(qn), and satis* *fies the following conditions: (a)For all P, Q S(qn) which contain z, if ff 2 Hom (P, Q) is such that ff(z) * *= z, then ff 2 Hom Fn(P, Q) if and only if ff 2 Hom Spin7(qn)(P, Q). (b)CFn(z) = FS(qn)(Spin7(qn)) as fusion systems over S(qn). (c)All involutions of S(qn) are Fn-conjugate. Furthermore, Fm Fn for m|n. The union of the Fn is thus a category FSol(q1 ) whose objects are the finite subgroups of S(q1 ). Proof.We apply Proposition 1.2, where p = 2, G = Spin7(qn), S = S(qn), Z = = Z(G); and U and CG(U) = H(qn) are as defined above. Also, = n Aut(H(qn)). Condition (a) in Proposition 1.2 (all noncentral involutions in G are conjugate* *) holds since all subgroups in E2 are conjugate (Proposition A.8), and condition (b) ho* *lds by definition of . Condition (c) holds since {fl 2 | fl(z) = z} = Inn(H(qn)).= AutNG(U)(H(qn)) by definition, since H(qn) = CG(U), and by Proposition 2.5(b). Condition (d) w* *as shown in Proposition 2.9, and condition (e) in Lemma 2.10. So by Proposition 1.* *2, Fn is a saturated fusion system, and CFn(Z) = FS(qn)(Spin7(qn)). The last statement is clear. 3.Linking systems and their automorphisms We next show the existence and uniqueness of centric linking systems associat* *ed to the FSol(q), and also construct certain automorphisms of these categories analo* *gous to the automorphisms _q of the group Spin7(qn). One more technical lemma about ele- mentary abelian subgroups, this time about their F-conjugacy classes, is first * *needed. Lemma 3.1. Set F = FSol(q). For each r 3, there is a unique F-conjugacy class* * of elementary abelian subgroups E S(q) of rank r. There are two F-conjugacy clas* *ses of rank four elementary abelian subgroups E S(q): one is the set C of subgro* *ups Spin7(q)-conjugate to E* = , while the other contains the other * *conjugacy class of type I subgroups as well as all type II subgroups. Furthermore, Aut F* *(E) = Ran Levi and Bob Oliver * * 19 Aut(E) for all elementary abelian subgroups E S(q) except when E has rank four and is not F-conjugate to E*, in which case Aut F(E) = {ff 2 Aut(E) | ff(xC(E)) = xC(E)}. Proof.By Lemma 2.8(d), the three subgroups E* = , E001= , E100= (where X is a generator of C(q)) represent the three Spin7(q)-conjugacy classes* * of rank four subgroups. Clearly, E100and E001are 1-conjugate, hence F-conjugate; and by Lemma 2.8(e), neither is 1-conjugate to E*. This proves that there are exactly* * two F-conjugacy classes of such subgroups. Since E* and E001both are of type I in Spin7(q), their Spin7(q)-automorphism * *groups contain all automorphisms which fix z (see Proposition A.8). By Lemma 2.8(e), z* * is fixed by all -automorphisms of E001, and so AutF (E001) is the group of all au* *tomor- phisms of E001which send z = xC(E001) to itself. On the other hand, E* contains automorphisms (induced by permuting the three coordinates of H) which permute t* *he three elements z, z1, zz1; and these together with AutSpin(E*) generate Aut(E*). It remains to deal with the subgroups of smaller rank. By Proposition A.8 ag* *ain, there is just one Spin7(q)-conjugacy class of elementary abelian subgroups of r* *ank one or two. There are two conjugacy classes of rank three subgroups, those of type * *I and those of type II. Since E100is of type II and E001of type I, all rank three sub* *groups of E001have type I, while some of the rank three subgroups of E100have type II.* * Since E001is F-conjugate to E100, this shows that some subgroup of rank three and typ* *e II is F-conjugate to a subgroup of type I, and hence all rank three subgroups are con* *jugate to each other. Finally, AutF (E) = Aut(E) whenver rk(E) 3 since any such grou* *p is F-conjugate to a subgroup of E* (and we have just seen that AutF(E*) = Aut(E*)). To simplify the notation, we now define FSpin(qn) def=FS(qn)(Spin7(qn)) for all 1 n 1: the fusion system of the group Spin7(qn) at the Sylow subgro* *up S(qn). By construction, this is a subcategory of FSol(qn). We write OSol(qn) = O(FSol(qn)) and OSpin(qn) = O(FSpin(qn)) for the corresponding orbit categories: both of these have as objects the subgr* *oups of S(qn), and have as morphism sets Mor OSol(qn)(P, Q) = Hom FSol(qn)(P, Q)= Inn(Q) Rep(P, Q) and MorOSpin(qn)(P, Q) = Hom FSpin(qn)(P, Q)= Inn(Q) . Let OcSol(qn) OSol(qn) and OcSpin(qn) OSpin(qn) be the centric orbit catego* *ries; i.e., the full subcategories whose objects are the FSol(qn)- or FSpin(qn)-centric sub* *groups of S(qn). (We will see shortly that these in fact have the same objects.) The obstructions to the existence and uniqueness of linking systems associate* *d to the fusion systems FSol(qn), and to the existence and uniqueness of certain automor* *phisms of those linking systems, lie in certain groups which were identified in [BLO2 * * ] and [BLO1 ]. It is these groups which are shown to vanish in the next lemma. 20 Construction of 2-local finite groups Lemma 3.2. Fix a prime power q, and let ZSol(q): OcSol(q) ----! Ab and ZSpin(q): OcSpin(q) ----! Ab be the functors Z(P ) = Z(P ). Then for all i 0, lim-i(ZSol(q)) = 0 = lim-i(ZSpin(q)). OcSol(q) OcSpin(q) Proof.Set F = FSol(q) for short. Let P1, . .,.Pk be F-conjugacy class represent* *atives for all F-centric subgroups Pi S(q), arranged such that |Pi| |Pj| for i j* *. For each i, let Zi ZSol(q) be the subfunctor defined by setting Zi(P ) = ZSol(q)(P* * ) if P is conjugate to Pj for some j i and Zi(P ) = 0 otherwise. We thus have a filt* *ration 0 = Z0 Z1 . . .Zk = ZSol(q) of ZSol(q) by subfunctors, with the property that for each i, the quotient func* *tor Zi=Zi-1vanishes except on the conjugacy class of Pi (and such that (Zi=Zi-1)(Pi* *) = ZSol(q)(Pi)). By [BLO2 , Proposition 3.2], lim-*(Zi=Zi-1) ~= *(Out F(Pi); Z(Pi)) for each i. Here, *( ; M) are certain graded groups, define in [JMO , x5] for* * all finite groups and all finite Z(p)[ ]-modules M. We claim that *(Out F(Pi); Z(Pi)) * *= 0 except when Pi= S(q) or S0(q) (see Definition 2.6). Fix an F-centric subgroup P S(q). For each j 1, let j(Z(P )) = {g 2 j Z(P ) | g2 = 1}, and set E = 1(Z(P )) _ the 2-torsion in the center of P . Fo* *r each j j 1, let j(Z(P )) = {g 2 Z(P ) | g2 = 1}, and set E = 1(Z(P )) _ the 2-tors* *ion in the center of P . We can assume E is fully centralized in F (otherwise replace * *P and E by appropriate subgroups in the same F-conjugacy classes). Assume first that Q def=CS(q)(E) P , and hence that NQ(P ) P . Then any x 2 NQ(P )r P centralizes E = 1(Z(P )). Hence for each j, x acts trivially * *on j(Z(P ))= j-1(Z(P )), since multiplication by pj-1sends this group NQ(P )=P -l* *inearly and monomorphically to E. Since cx is a nontrivial element of Out F(P ) of p-p* *ower order, *(Out F(P ); j(Z(P ))= j-1(Z(P ))) = 0 for all j 1 by [JMO , Proposition 5.5], and thus *(Out F(P ); Z(P )) = 0. Now assume that P = CS(q)(E) = P , the centralizer in S(q) of a fully F-centr* *alized elementary abelian subgroup. Since there is a unique conjugacy class of elemen* *tary abelian subgroup of any rank 3, CS(q)(E) always contains a subgroup C42, and * *hence P contains a subgroup C42which is self centralizing by Proposition A.8(a). This* * shows that Z(P ) is elementary abelian, and hence that Z(P ) = E. We can assume P is fully normalized in F, so Aut S(q)(P ) 2 Syl2(Aut F(P )) by condition (I) in the definition of a saturated fusion system. Since P = CS(* *q)(E) (and E = Z(P )), this shows that Ker OutF(P ) ---! AutF(E) has odd order. Also, since E is fully centralized, any F-automorphism of E exte* *nds to an F-automorphism of P = CS(q)(E), and thus this restriction map between automo* *r- phism groups is onto. By [JMO , Proposition 6.1(i,iii)], it now follows that i(Out F(P ); Z(P )) ~= i(Aut F(E); E). (1) Ran Levi and Bob Oliver * * 21 By Lemma 3.1, AutF (E) = Aut(E), except when E lies in one certain F-conjugacy class of subgroups E ~=C42; and in this case P = E and AutF (E) is the group of* * auto- morphisms fixing the element xC(E). In this last (exceptional) case, O2(Aut F(E* *)) 6= 1 (the subgroup of elements which are the identity on E=), so *(Out F(P ); Z(P )) = *(Aut F(E); E) = 0 (2) by [JMO , Proposition 6.1(ii)]. Otherwise, when AutF (E) = Aut(E), by [JMO , * *Propo- sition 6.3] we have 8 >: 0 otherwise. By points (1), (2), and (3), the groups *(Out F(P ); Z(P )) vanish except in t* *he two cases E = or E = U, and these correspond to P = S(q) or P = NS(q)(U) = S0(q* *). We can assume that Pk = S(q) and Pk-1 = S0(q). We have now shown that lim-*(Zk-2) = 0, and thus that ZSol(q) has the same higher limits as Zk=Zk-2. H* *ence lim-j(ZSol(q)) = 0 for all j 2, and there is an exact sequence 0 ---! lim-0(ZSol(q)) ----! lim-0(Zk=Zk-1)----! lim1(Zk-1=Zk-2) ~=Z=2 - ~=Z=2 ----! lim-1(ZSol(q)) ---! 0. One easily calculates that lim-0(ZSol(q)) = 0, and hence we also get lim-1(ZSol* *(q)) = 0. The proof that lim-i(ZSpin(q)) = 0 for all i 1 is similar, but simpler. If * *F = FSpin(q), then for any F-centric subgroup P S(q), there is an element x 2 NS(P )r P such that [x, P ] = , and cx is a nontrivial element of O2(Out F(P )). Thus *(Out F(P ); Z(P )) = 0 for all such P by [JMO , Proposition 6.1(ii)] again. We are now ready to construct classifying spaces BSol(q) for these fusion sys* *tems FSol(q). The following proposition finishes the proof of Theorem 2.1, and also * *contains additional information about the spaces BSol(q). To simplify notation, we write LcSpin(qn) = LcS(qn)(Spin7(qn)) (n 1) to den* *ote the centric linking system for the group Spin7(qn). The field automorphism (x 7! x* *q) induces an automorphism of Spin7(qn) which sends S(qn) to itself; and this in t* *urn induces automorphisms _qF= _qF(Sol), _qF(Spin), and _qL(Spin) of the fusion sys* *tems FSol(qn) FSpin(qn) and of the linking system LcSpin(qn). Proposition 3.3. Fix an odd prime q, and n 1. Let S = S(qn) 2 Syl2(Spin7(qn))* * be as defined above. Let z 2 Z(Spin7(qn)) be the central element of order 2. Then * *there is a centric linking system L = LcSol(qn) ---i--!FSol(qn) associated to the saturated fusion system F def=FSol(qn) over S, which has the * *following additional properties. (a)A subgroup P S is F-centric if and only if it is FSpin(qn)-centric. 22 Construction of 2-local finite groups (b)LcSol(qn) contains LcSpin(qn) as a subcategory, in such a way that ß|LcSpin(* *qn)is the usual projection to FcSpin(qn), and that the distinguished monomorphisms ffiP P ---! AutL(P ) for L = LcSol(qn) are the same as those for LcSpin(qn). (c)Each automorphism of LcSpin(qn) which covers the identity on FcSpin(qn) exte* *nds to an automorphism of LcSol(qn) which covers the identity on FcSol(qn). Further* *more, such an extension is unique up to composition with the functor Cz: LcSol(qn) -----! LcSol(qn) which is the identity on objects and sends ff 2 Mor LcSol(qn)(P, Q) to bzOff* * Obz-1 (öc njugation by z"). (d)There is a unique automorphism _qL2 Aut (LcSol(qn)) which covers the automor- phism of FSol(qn) induced by the field automorphism (x 7! xq), which extends* * the automorphism of LcSpin(qn) induced by the field automorphism, and which is t* *he identity on ß-1(FSol(q)). Proof.By Proposition 2.11, F = FSol(qn) is a saturated fusion system over S = S* *(qn) 2 Syl2(Spin7(qn)), with the property that CF (z) = FSpin(qn). Point (a) follows * *as a special case of [BLO2 , Proposition 2.5(a)]. Since lim-i(ZSol(qn))= 0 for i = 2, 3 by Lemma 3.2, there is by [BLO2 , Pr* *opo- OcSol(qn) sition 3.1] a centric linking system L = LcSol(qn) associated to F, which is un* *ique up to isomorphism (an isomorphism which commutes with the projection to FSol(qn) a* *nd with the distinguished monomorphisms). Furthermore, ß-1(FSpin(qn)) is a linking* * sys- tem associated to FSpin(qn), such a linking system is unique up to isomorphism * *since lim-2(ZSpin(qn)) = 0 (Lemma 3.2 again), and this proves (b). (c) By [BLO1 , Theorem 6.2] (more precisely, by the same proof as that used in [BLO1 ]), the vanishing of lim-i(ZSol(qn)) for i = 1, 2 (Lemma 3.2) shows that* * each automorphism of F = FSol(qn) lifts to an automorphism of L, which is unique up to a natural isomorphism of functors; and any such natural isomorphism sends ea* *ch object P S to a isomorphism bgfor some g 2 Z(P ). Similarly, the vanishing * *of lim-i(ZSpin(qn)) for i = 1, 2 shows that each automorphism of FSpin(qn) lifts t* *o an automorphism of LcSpin(qn), also unique up to a natural isomorphism of functors* *. Since LcSol(qn) and LcSpin(qn) have the same objects by (a), this shows that each aut* *omorphism of LcSpin(qn) which covers the identity on FcSpin(qn) extends to a unique autom* *orphism of LcSol(qn) which covers the identity on FSol(qn). It remains to show, for any 2 Aut(LcSol(qn)) which covers the identity on F* *cSol(qn) and such that |LcSpin(qn)= Id, that is the identity or conjugation by z. We* * have already noted that must be naturally isomorphic to the identity; i.e., that t* *here are elements fl(P ) 2 Z(P ), for all P in LcSol(qn), such that (ff) = fl(Q) Off Ofl(P )-1 for all ff 2 Mor LcSol(qn)(P, Q), all P,* * Q. Since is the identity on LcSpin(qn), the only possibilities are fl(P ) = 1 fo* *r all P (hence = Id), or fl(P ) = z for all P (hence is conjugation by z). (d) Now consider the automorphism _qF2 Aut(FSol(qn)) induced by the field auto- morphism (x 7! xq) of Fqn. We have just seen that this lifts to an automorphism* * _qL of LcSol(qn), which is unique up to natural isomorphism of functors. The restri* *ction of Ran Levi and Bob Oliver * * 23 q q _L to LcSpin(qn), and the automorphism _L(Spin) of LcSpin(qn) induced directly * *by the field automorphism, are two liftings of _qF|FSpin(qn), and hence differ by a na* *tural iso- morphism of functors which extends to a natural isomorphism of functors on LcSo* *l(qn). Upon composing with this natural isomorphism, we can thus assume that _qLdoes restrict to the automorphism of LcSpin(qn) induced by the field automorphism. Now consider the action of _qLon AutL(S0(q)), which by assumption is the iden* *tity on Aut LcSpin(q)(S0(q)), and in particular on ffi(S0(q)) itself. Thus, with re* *spect to the extension 1 ---! S0(q) ----! AutL(S0(q)) ----! 3 ---! 1, _qLis the identity on the kernel and on the quotient, and hence is described by* * a cocycle j 2 Z1( 3; Z(S0(q))) ~=Z1( 3; (Z=2)2). Since H1( 3; (Z=2)2) = 0, j must be a coboundary, and thus the action of _qLon AutL(S0(q)) is conjugation by an element of Z(S0(q)). Since it is the identity* * on AutLcSpin(q)(S0(q)), it must be conjugation by 1 or z. If it is conjugation by * *z, then we can replace _qL(on the whole category L) by its composite with z; i.e., by its * *composite with the functor which is the identity on objects and sends ff 2 Mor L(P, Q) to* * bzOff Obz. In this way, we can assume that _qLis the identity on AutL(S0(q)). By constru* *ction, every morphism in FSol(q) is a composite of morphisms in FSpin(q) and restricti* *ons of automorphisms in FSol(q) of S0(q). Since _qLis the identity on ß-1(FSpin(q))* *, this shows that it is the identity on ß-1(FSol(q)). It remains to check the uniqueness of _qL. If _0 is another functor with the* * same properties, then by (e), (_0)-1 O_qLis either the identity or conjugation by z;* * and the latter is not possible since conjugation by z is not the identity on ß-1(FSol(q* *)). This finishes the construction of the classifying spaces BSol(q) = |LSol(q)|^* *2for the fusion systems constructed in Section 2. We end the section with an explanatio* *n of why these are not the fusion systems of finite groups. Proposition 3.4. For any odd prime power q, there is no finite group G whose fu* *sion system is isomorphic to that of FSol(q). Proof.Let G be a finite group, fix S 2 Syl2(G), and assume that S ~= S(q) 2 Syl2(Spin7(q)), and that the fusion system FS(G) satisfies conditions (a) and (* *b) in Theorem 2.1. In particular, all involutions in G are conjugate, and the centra* *lizer of any involution z 2 G has the fusion system of Spin7(q). When q 3 (mod 8), Solomon showed [So, Theorem 3.2] that there is no finite group whose fusion sys* *tem has these properties. When q 1 (mod 8), he showed (in the same theorem) that there is no such G such that bH def=CG(z)=O20(CG(z)) is isomorphic to a subgrou* *p of Aut(Spin7(q)) which contains Spin7(q) with odd index. (Here, O20(-) means larg* *est odd order normal subgroup.) Let G be a finite group whose fusion system is isomorphic to FSol(q), and aga* *in 0 set Hb def=CG(z)=O20(CG(z)) for some involution z 2 G. Set H = O2 (Hb=): t* *he smallest normal subgroup of Hb= of odd index. Then H has the fusion system * *of 7(q) ~=Spin7(q)=Z(Spin7(q)). We will show that H ~= 7(q0) for some odd prime p* *ower 0 0 q0. It then follows that O2 (Hb) ~=Spin7(q ), thus contradicting Solomon's theo* *rem and proving our claim. The following "classification freeä rgument for proving that H ~= 7(q0) for * *some q0 was explained to us by Solomon. We refer to the appendix for general results ab* *out 24 Construction of 2-local finite groups the groups Spinn(q) and n(q). Fix S 2 Syl2(H). Thus S is isomorphic to a Sylow 2-subgroup of 7(q), and has the same fusion. 0 We first claim that H must be simple. By definition (H = O2 (Hb=)), H has no proper normal subgroup of odd index, and H has no proper normal subgroup of odd order since any such subgroup would lift to an odd order normal subgroup of bH= CG(z)=O20(CG(z)). Hence for any proper normal subgroup N C H, Q def=N \ S is a proper normal subgroup of S, which is strongly closed in S with respect to* * H in the sense that no element of Q can be H-conjugate to an element of Sr Q. Usi* *ng Lemma A.4(a), one checks that the group 7(q) contains three conjugacy classes * *of involutions, classified by the dimension of their (-1)-eigenspace. It is not ha* *rd to see (by taking products) that any subgroup of S which contains all involutions in o* *ne of these conjugacy classes contains all involutions in the other two classes as* * well. Furthermore, S is generated by the set of all of its involutions, and this show* *s that there are no proper subgroups which are strongly closed in S with respect to H.* * Since we have already seen that the intersection with S of any proper normal subgroup* * of H would have to be such a subgroup, this shows that H is simple. Fix an isomorphism ' 0 S -------!~S 2 Syl2( 7(q)) = which preserves fusion. Choose x0 2 S0 whose (-1)-eigenspace is 4-dimensional, * *and such that is fully centralized in FS0( 7(q)). Then CO7(q)(x0) ~=O+4(q) x O3(q) by Lemma A.4(c). Since +4(q) O+4(q) and 3(q) O3(q) both have index 4, C 7(q)(x0) is isomorphic to a subgroup of O+4(q) x O3(q) of index 4, and contai* *ns a normal subgroup K0~= +4(q) x 3(q) of index 4. Since is fully centralized,* * CS0(x0) is a Sylow 2-subgroup of C 7(q)(x0), and hence S00def=S0\ K0 is a Sylow 2-subgr* *oup of K0. Set x = '-1(x0) 2 S. Since S ~=S0 have the same fusion in H and 7(q), CS(x) * *~= CS0(x0) have the same fusion in CH (x) and C 7(q)(x0). Hence H1(CH (x); Z(2)) * * ~= H1(C 7(q)(x0); Z(2)) (homology is determined by fusion), both have order 4, and* * thus CH (x) also has a unique normal subgroup K C H of index 4. Set S0 = K \ S. Thus '(S0) = S00, and using Alperin's fusion theorem one can show that this isomorph* *ism is fusion preserving with respect to the inclusions of Sylow subgroups S0 K a* *nd S00 K0. Using the isomorphisms of Proposition A.5: +4(q) ~=SL2(q) xSL2(q) and 3(q) ~=P SL2(q), we can write K0= K01xK02, where K01~=SL2(q) and K02~=SL2(q) x P SL2(q). Set S0i= S0\K0i2 Syl2(K0i); thus S00= S01xS02. Set Si= '-1(S0i), so that S0 = S* *1xS2 is normal of index 4 in CS(x). The fusion system of K thus splits as a central * *product of fusion systems, one of which is isomorphic to the fusion system of SL2(q). We now apply a theorem of Goldschmidt, which says very roughly that under the* *se conditions, the group K also splits as a central product. To make this more pre* *cise, let Kibe the normal closure of Siin K C CH (x). By [Gd , Corollary A2], since S* *1 and S2 are strongly closed in S0 with respect to K, [K1, K2] .O20(K). Ran Levi and Bob Oliver * * 25 Using this, it is not hard to check that Si 2 Syl2(Ki). Thus K1 has same fusio* *n as SL2(q) and is subnormal in CH (x) (K1 C K C CH (x)), and an argument similar to that used above to prove the simplicity of H shows that K1=(.O20(K1)) is sim* *ple. Hence K1 is a 2-component of CH (x) in the sense described by Aschbacher in [As* *1 ]. By [As1 , Corollary III], this implies that H must be isomorphic to a Chevalley* * group of odd characteristic, or to M11. It is now straightforward to check that among* * these groups, the only possibility is that H ~= 7(q0) for some odd prime power q0. 4.Relation with the Dwyer-Wilkerson space We now want to examine the relation between the spaces BSol(q) which we have * *just constructed, and the space BDI(4) constructed by Dwyer and Wilkerson in [DW1 ]. Recall that this is a 2-complete space characterized by the property that its c* *oho- mology is the Dickson algebra in four variables over F2; i.e., the ring of inva* *riants F2[x1, x2, x3, x4]GL4(2). We show, for any odd prime power q, that BDI(4) is ho* *motopy equivalent to the 2-completion of the union of the spaces BSol(qn), and that BS* *ol(q) is homotopy equivalent to the homotopy fixed point set of an Adams map from BDI(4) to itself. We would like to define an infinite "linking system" LcSol(q1 ) as the union * *of the finite categories LcSol(qn), and then set BSol(q1 ) = |LcSol(q1 )|^2. The diffi* *culty with this approach is that a subgroup which is centric in the fusion system FSol(qm ) nee* *d not be centric in a larger fusion system FSol(qn) (for m|n). To get around this proble* *m, we define LccSol(qn) LcSol(qn) to be the full subcategory whose objects are thos* *e subgroups of S(qn) which are FSol(q1 )-centric; or equivalently FSol(qk)-centric for all * *k 2 nZ. Similarly, we define LccSpin(qn) to be the full subcategory of LcSpin(qn) whose* * objects are those subgroups of S(qn) which are FSpin(q1 )-centric. We can then define LcSo* *l(q1 ) and LcSpin(q1 ) to be the unions of these categories. For these definitions to be useful, we must first show that |LccSol(qn)|^2has* * the same homotopy type as |LcSol(qn)|^2. This is done in the following lemma. Lemma 4.1. For any odd prime power q and any n 1, the inclusions |LccSol(qn)|^2 |LcSol(qn)|^2 and |LccSpin(qn)|^2 |LcSpin(qn)|* *^2 are homotopy equivalences. Proof.It clearly suffices to show this when n = 1. Recall, for a fusion system F over a p-group S, that a subgroup P S is F-ra* *dical if OutF (P ) is p-reduced; i.e., if Op(Out F(P )) = 1. We will show that all FSol(q)-centric FSol(q)-radical subgroups of S(q) are also FSol(q1 )-cen* *tric(1) and similarly all FSpin(q)-centric FSpin(q)-radical subgroups of S(q) are also FSpin(q1()-ce* *ntric.2) In other words, (1) says that for each P S(q) which is an object of LcSol(q) * *but not of LccSol(q), O2 Out FSol(q)(P ) 6= 1. By [JMO , Proposition 6.1(ii)], this i* *mplies that *(Out FSol(q)(P ); H*(BP ; F2)) = 0. 26 Construction of 2-local finite groups Hence by [BLO2 , Propositions 3.2 & 2.2] (and the spectral sequence for a homo* *topy colimit), the inclusion LccSol(q) LcSol(q) induces an isomorphism c ~= * cc H* |LSol(q)|; F2 ------! H |LSol(q)|; F2 , and thus |LccSol(q)|^2' |LcSol(q)|^2. The proof that |LccSpin(q)|^2' |LcSpin(q* *)|^2is similar, using (2). Point (2) is shown in Proposition A.12, so it remains only to prove (1). Set* * F = FSol(q), and set Fk = FSol(qk) for all 1 k 1. Let E Z(P ) be the 2-torsio* *n in the center of P , so that P CS(q)(E). Set 8 >> if rk(E) = 1 >< if rk(E) = 2 E0= >>if rk(E) = 3 >: E if rk(E) = 4 in the notation of Definition 2.6. In all cases, E is F-conjugate to E0 by Lem* *ma 3.1. We claim that E0 is fully centralized in Fk for all k < 1. This is clear* * when rk(E0) = 1 (E0 = Z(S(qk))), follows from Proposition 2.5(a) when rk(E0) = 2, and from Proposition A.8(a) (all rank 4 subgroups are self centralizing) when rk(E0* *) = 4. If rk(E0) = 3, then by Proposition A.8(d), the centralizer in Spin7(qk) (hence * *in S(qk)) of any rank 3 subgroup has an abelian subgroup of index 2; and using this (toge* *ther with the construction of S(qk) in Definition 2.6), one sees that E0 is fully ce* *ntralized in Fk. _ If E0 6= E, choose ' 2 Hom F (E, S(q)) such that '(E) = E0; then ' extends to ' 2 Hom F(CS(q)(E), S(q)) by condition_(II) in the definition of a saturated f* *usion system, and we can replace P by '(P ) and E by '(E). We can thus assume that E is fully centralized in Fk for each k < 1. So by [BLO2 , Proposition 2.5(a)* *], P is Fk-centric if and only if it is CFk(E)-centric; and this also holds when k =* * 1. Furthermore, since OutCF(E)(P ) C OutF (P ), O2(Out CF(E)(P )) is a normal 2-su* *bgroup of OutF (P ), and thus O2 Out CF(E)(P ) O2(Out F(P )). Hence P is CF (E)-radical if it is F-radical. So it remains to show that all CF (E)-centric CF (E)-radical subgroups of S(q) are also CF1 (E)-centric* *.(3) If rk(E) = 1, then CF (E) = FSpin(q) and CF1 (E) = FSpin(q1 ), and (3) follow* *s from (2). If rk(E) = 4, then P = E = CS(q1 )(E) by Proposition A.8(a), so P is F1 -c* *entric, and the result is clear. If rk(E) = 3, then by Proposition A.8(d), CF (E) CF1 (E) are the fusion sys* *tems of a pair of semidirect products Ao C2 A1 oC2, where A A1 are abelian and * *C2 acts on A1 by inversion. Also, E is the full 2-torsion subgroup of A1 , since o* *therwise rk(A1 ) > 3 would imply A1 oC2 Spin7(q1 ) contains a subgroup C52(contradicti* *ng Proposition A.8). If P A, then either Out CF(E)(P ) has order 2, which contra* *dicts the assumption that P is radical; or P is elementary abelian and Out CF(E)(P ) * *= 1, in which case P Z(Ao C2) is not centric. Thus P A; P \ A E contains all 2-torsion in A1 , and hence P is centric in A1 oC2. If rk(E) = 2, then by Proposition 2.5(a), CF1 (E) and CF (E) are the fusion s* *ystems of the groups H(q1 ) ~=SL2(q1 )3={ (I, I, I)} (4) Ran Levi and Bob Oliver * * 27 and H(q) = H(q1 ) \ Spin7(q) H0(q) def=SL2(q)3={ (I, I, I)}. If P S(q) is centric and radical in the fusion system of H(q), then by Lemma * *A.11(c), its intersection with H0(q) ~=SL2(q)3={ (I, I, I)} is centric and radical in th* *e fusion system of that group. So by Lemma A.11(a,f), P \ H0(q) ~=(P1 x P2 x P3)={ (I, I, I)} (5) for some Piwhich are centric and radical in the fusion system of SL2(q). Since * *the Sylow 2-subgroups of SL2(q) are quaternion [Go , Theorem 2.8.3], the Pimust be nonabe* *lian and quaternion, so each Pi={ I} is centric in P SL2(q1 ). Hence P \ H0(q) is ce* *ntric in SL2(q)3={ (I, I, I)} by (5), and so P is centric in H(q1 ) by (4). We would like to be able to regard BSpin7(q) as a subcomplex of BSol(q), but * *there is no simple natural way to do so. Instead, we set BSpin07(q) = |LccSpin(q)|^2 |LccSol(q)|^2 BSol(q); then BSpin07(q) ' BSpin7(q)^2by [BLO1 , Proposition 1.1] and Lemma 4.1. Also, * *we write BSol0(q) = |LccSol(q)|^2 BSol(q) def=|LcSol(q)|^2 to denote the subcomplex shown in Lemma 4.1 to be equivalent to BSol(q); and set BSpin07(q1 ) = |LcSpin(q1 )|^2. From now on, when we talk about the inclus* *ion of BSpin7(q) into BSol(q), as long as it need only be well defined up to homotopy,* * we mean the composite BSpin7(q) ' BSpin07(q) BSol0(q) (for some choice of homotopy equivalence). Similarly, if we talk about the inc* *lu- sion of BSol(qm ) into BSol(qn) (for m|n) where it need only be defined up to h* *omo- topy, we mean these spaces identified with their equivalent subcomplexes BSol0(* *qm ) BSol0(qn). Lemma 4.2. Let q be any odd prime. Then for all n 1, H*(BSol(qn); F2)___! H*(BH(qn); F2)C3 | | | | (1) # # H*(BSpin7(qn); F2)__! H*(BH(qn); F2) (with all maps induced by inclusions of groups or spaces) is a pullback square. Proof.By [BLO2 , Theorem B], H*(BSol(qn); F2) is the ring of elements in the c* *oho- mology of S(qn) which are stable relative to the fusion. By the construction in* * Section 2, the fusion in Sol(qn) is generated by that in Spin7(qn), together with the p* *ermu- tation action of C3 on the subgroup H(qn) Spin7(qn), and hence (1) is a pullb* *ack square. Proposition 4.3. For each odd prime q, there is a category LcSol(q1 ), together* * with a functor ß :LcSol(q1 ) ------! FSol(q1 ), such that the following hold: (a)For each n 1, ß-1(FSol(qn)) ~=LccSol(qn). 28 Construction of 2-local finite groups (b)There is a homotopy equivalence '' BSol(q1 ) def=|LcSol(q1 )|^2-------! BDI(4) ' such that the following square commutes up to homotopy ffi(q1 ) 1 BSpin07(q1 )^2_____! BSol(q ) | | ''0'| '''| (1) # b # ffi BSpin(7)^2________! BDI(4) . Here, j0 is the homotopy equivalence_of [FM ], induced by some fixed choice * *of em- bedding of the Witt vectors for Fq into C, while ffi(q1 ) is the union of th* *e inclusions |LccSpin(qn)|^2 |LccSol(qn)|^2, and bffiis the inclusion arising from the c* *onstruction of BDI(4) in [DW1 ]. Furthermore, there is an automorphism _qL2 Aut(LcSol(q1 )) of categories which * *satis- fies the conditions: (c)the restriction of _qLto each subcategory LccSol(qn) is equal to the restric* *tion of _qL2 Aut (LcSol(qn)) as defined in Proposition 3.3(d); (d)_qLcovers the automorphism _qFof FSol(q1 ) induced by the field automorphism (x 7! xq); and (e)for each n, (_qL)n fixes LccSol(qn). Proof.By Proposition 2.11, the inclusions Spin7(qm ) Spin7(qn) for all m|n in* *duce inclusions of fusion systems FSol(qm ) FSol(qn). Since the restriction of a * *linking system over FccSol(qn) is a linking system over FccSol(qm ), the uniqueness of * *linking systems (Proposition 3.3) implies that we get inclusions LccSol(qm ) LccSol(qn). We d* *efine LcSol(q1 ) to be the union of the finite categories LccSol(qn). (More precisely, fix a se* *quence of positive integers n1|n2|n3| . .s.uch that every positive integer divides some n* *i, and set 1[ LcSol(q1 ) = LccSol(qni). i=1 Then by uniqueness again, we can identify LccSol(qn) for each n with the approp* *riate subcategory.) Let ß :LcSol(q1 ) --! FSol(q1 ) be the union of the projections from LccSol(q* *ni) to FSol(qni) FSol(q1 ). Condition (a) is clearly satisfied. Also, using Proposit* *ion 3.3(d) and Lemma 4.1, we see that there is an automorphism _qLof LcSol(q1 ) which sati* *sfies conditions (c,d,e) above. (Note that by the fusion theorem as shown in [BLO2 ,* * The- orem A.10], morphisms in LcSol(qn) are generated by those between radical subgr* *oups, and hence by those in LccSol(qn).) It remains only to show that |LcSol(q1 )|^2' BDI(4), and to show that square * *(1) commutes. The space BDI(4) is 2-complete by its construction in [DW1 ]. By Lem* *ma 4.1, c n * n H*(BSol(q1 ); F2) ~=lim-H* |LSol(q )|; F2 = lim-H BSol(q ); F2 . n n Ran Levi and Bob Oliver * * 29 Hence by Lemma 4.2 (and since the inclusions BSpin7(qn) --! BSol(qn) commute wi* *th the maps induced by inclusions of fields Fqm Fqn), there is a pullback square H*(BSol(q1 ); F2)_! H*(BH(q1 ); F2)C3 | | | | (2) # # H*(BSpin7(q1 ); F2)_! H*(BH(q1 ); F2) . Also, by [FM , Theorem 1.4], there are maps 3 BSpin7(q1 ) ----! BSpin(7) and BH(q1 ) ----! B SU(2) ={ (I, I, I)} which induce isomorphisms of F2-cohomology, and hence homotopy equivalences aft* *er 2-completion. So by Propositions 4.7 and 4.9 (or more directly by the computati* *ons in [DW1 , x3]), the pullback of the above square is the ring of Dickson invari* *ants in the polynomial algebra H*(BC42; F2), and thus isomorphic to H*(BDI(4); F2). Point (b), including the commutativity of (1), now follows from the following* * lemma. Lemma 4.4. Let X be a 2-complete space such that H*(X; F2) is the Dickson alge- f bra in 4 variables. Assume further that there is a map BSpin(7) --! X such that H*(f|BC42; F2) is the inclusion of the Dickson invariants in the polynomial alg* *ebra H*(BC42; F2). Then X ' BDI(4). More precisely, there is a homotopy equivalence between these spaces such that the composite f BSpin(7) ------! X ' BDI(4) is the inclusion arising from the construction in [DW1 ]. Proof.In fact, Notbohm [Nb , Theorem 1.2] has proven that the lemma holds even without the assumption about BSpin(7) (but with the more precise assumption that H*(X; F2) is isomorphic as an algebra over the Steenrod algebra to the Dickson * *alge- bra). The result as stated above is much more elementary (and also implicit in * *[DW1 ]), so we sketch the proof here. Since H*(X; F2) is a polynomial algebra, H*( X; F2) is isomorphic as a graded vector space to an exterior algebra on the same number of variables, and in par* *ticular is finite. Hence X is a 2-compact group. By [DW3 , Theorem 8.1] (the central* *izer decomposition for a p-compact group), there is an F2-homology equivalence hocolim-----!(ff) ---'---!X. A Here, A is the category of pairs (V, '), where V is a nontrivial elementary ab* *elian 2-group, and ' : BV --! X makes H*(BV ; F2) into a finitely generated module o* *ver H*(X; F2) (see [DW2 , Proposition 9.11]). Morphisms in A are defined by letti* *ng Mor A((V, '), (V 0, '0)) be the set of monomorphisms V --! V 0of groups which * *make the obvious triangle commute up to homotopy. Also, ff: Aop--! Top is the functorff(V, ') = Map (BV, X)'. By [DW1 , Lemma 1.6(1)] and [La , Th'eor`eme 0.4], A is equivalent to the cate* *gory of elementary abelian 2-groups E with 1 rk(E) 4, whose morphisms consist of all ' group monomorphisms. Also, if BC2 --! X is the restriction of f to any subgroup C2 Spin(7), then in the notation of Lannes, TC2(H*(X; F2); '*) ~=H*(BSpin(7); F2) 30 Construction of 2-local finite groups by [DW1 , Lemmas 16.(3), 3.10 & 3.11], and hence H*(Map (BC2, X)'; F2) ~=H*(BSpin(7); F2) by Lannes [La , Th'eor`eme 3.2.1]. This shows that i j Map (BC2, X)' ^2' BSpin(7)^2, and thus that the centralizers of other elementary abelian 2-groups are the sam* *e as their centralizers in BSpin(7)^2. In other words, ff is equivalent in the homotopy c* *ategory to the diagram used in [DW1 ] to define BDI(4). By [DW1 , Proposition 7.7] (a* *nd the remarks in its proof), this homotopy functor has a unique homotopy lifting to s* *paces. So by definition of BDI(4), ^ X ' hocolim-----!(ff) 2 ' BDI(4). A Set B_q def=|_qL|, a self homotopy equivalence of BSol(q1 ) ' BDI(4). By cons* *truc- tion, the restriction of B_q to the maximal torus of BSol(q1 ) is the map induc* *ed by x 7! xq, and hence this is an Ä dams mapä s defined by Notbohm [Nb ]. In fact, by [Nb , Theorem 3.5], there is an Adams map from BDI(4) to itself, unique up to homotopy, of degree any 2-adic unit. Following Benson [Be ], we define BDI4(q) for any odd prime power q to be the homotopy fixed point set of the Z-action on BSol(q1 ) ' BDI(4) induced by the Adams map B_q. By öh motopy fixed point set" in this situation, we mean that the following square is a homotopy pullback: BDI4(q) ____________! BSol(q1 ) | | | | # # (Id,B_q) 1 1 BSol(q1 )______!BSol(q ) x BSol(q ). The actual pullback of this square is the subspace BSol(q) of elements fixed by* * B_q, and we thus have a natural map BSol(q) -ffi0-!BDI4(q). Theorem 4.5. For any odd prime power q, the natural map BSol(q) ----ffi0---!BDI4(q) ' is a homotopy equivalence. Proof.Since BDI(4) is simply connected, the square used to define BDI4(q) remai* *ns a homotopy pullback square after 2-completion by [BK , II.5.3]. Thus BDI4(q) i* *s 2- complete. Also, BSol(q) def=|LcSol(q)|^2is 2-complete since |LcSol(q)| is 2-go* *od [BLO2 , Proposition 1.11], and hence it suffices to prove that the map between these sp* *aces is an F2-cohomology equivalence. By Lemma 4.2, this means showing that the followi* *ng commutative square is a pullback square: H*(BDI4(q); F2) ___! H*(BH(q); F2)C3 | | | | (1) # # H*(BSpin7(q); F2)___! H*(BH(q); F2) . Here, the maps are induced by the composite BSpin7(q) ' BSpin07(q)^2 BSol(q) ------! BDI4(q) Ran Levi and Bob Oliver * * 31 and its restriction to BH(q). Also, by Proposition 4.3(b), the following diagra* *m com- mutes up to homotopy: incl 1 ''0 BSpin7(q) ___! BSpin7(q )____! BSpin(7) | | | ffi(q)| ffi(q1|) bffi| (2) # # # incl 1 '' BSol(q) _____! BSol(q )_____! BDI(4) By [Fr, Theorem 12.2], together with [FM_, x1], for any connected reductive L* *ie group G and any algebraic epimorphism _ on G(F q) with finite fixed subgroup, there i* *s a homotopy pullback square _ incl _ B(G(F q)_)^2_________!BG(F q)^2 incl#| #| (3) _ (Id,B_) _ _ BG(F q)^2______!BG(F q)^2x BG(F q)^2. We need to apply this when G = Spin7or G = H = (SL2)3={ (I, I, I)}. In particul* *ar, if _ =__q is the automorphism induced by the_field automorphism_(x 7! xq), then Spin7(F q)_ = Spin7(q) by Lemma A.3, and H(F q)_ = H(q) def=H(F q) \ Spin7(q). * *We thus get a description of BSpin7(q) and BH(q) as homotopy pullbacks. _ By [FM , Theorem 1.4], BG(F q)^2' BG(C)^2. Also, we can replace the complex __ def Lie groups Spin7(C) and H(C) by maximal compact subgroups Spin(7) and H = SU(2)3={ (I, I, I)}, since these have the same homotopy type. _ If we set R = H*(BG(F q); F2) ~=H*(BG(C); F2), then there are Eilenberg-Moore spectral sequences _ E2 = Tor*R Rop(R, R) =) H*(B(G(F q)_); F2); where the (R Rop)-module structure on R is defined by setting (a b).x = a.x.B_(* *b). When G = Spin7or H, then R is a polynomial algebra by Proposition 4.7 and the a* *bove remarks, and B_ acts on R via the identity. The above spectral sequence thus sa* *tisfies the hypotheses of [Sm , Theorem II.3.1], and hence collapses. (Alternatively, n* *ote that in this case, E2 is generated multiplicatively by E0,*2and E-1,*2by (5) below.)* * Similarly, when R = H*(BDI(4); F2), there is an analogous spectral sequence which converge* *s to H*(BDI4(q); F2), and which collapses for the same reason. By the above remarks,* * these spectral sequences are natural with respect to the inclusions BH(-) BSpin7(-)* *, and (using the naturality of _q shown in Proposition 3.3(d)) of BSpin7(-) into BSol* *(-) or BDI(4). To simplify the notation, we now write __ A def=H*(BDI(4); F2), B def=H*(BSpin(7); F2), and C def=H*(H ; F2) to denote these cohomology rings. The Frobenius automorphism _q acts via the id* *entity on each of them. We claim that the square Tor*A Aop(A, A)__!Tor*C Cop(C, C)C3 | | | | (4) # # Tor*B Bop(B, B)___! Tor*C Cop(C, C) 32 Construction of 2-local finite groups is a pullback square. Once this has been shown, it then follows that in each d* *egree, square (1) has a finite filtration under which each quotient is a pullback squa* *re. Hence (1) itself is a pullback. For any commutative F2-algebra R, let R=F2 denote the R-module generated by elements dr for r 2 R with the relations dr = 0 if r 2 F2, d(r + s) = dr + ds and d(rs) = r.ds + s.dr. Let *R=F2denote the ring of Kähler differentials: the exterior algebra (over * *R) of R=F2= 1R=F2. When R is a polynomial algebra, there are natural identifications Tor*R Rop(R, R) ~=HH*(R; R) ~= *R=F2. (5) The first isomorphism holds for arbitrary algebras, and is shown, e.g., in [Wb * *, Lemma 9.1.3]. The second holds for smooth algebras over a field [Wb , Theorem 9.4.7]* * (and polynomial algebras are smooth as shown in [Wb , x9.3.1]). In particular, the * *iso- morphisms (5) hold for R = A, B, C, which are shown to be polynomial algebras in Proposition 4.7 below. Thus, square (4) is isomorphic to the square * C *A=F2__! C=F2 3 | | | | (6) # # *B=F2___! *C=F2, which is shown to be a pullback square in Propositions 4.7 and 4.9 below. It remains to prove that square (6) in the above proof is a pullback square. * *In what follows, we let Di(x1, . .,.xn) denote the i-th Dickson invariant in variables * *x1, . .,.xn. This is the (2n- 2n-i)-th symmetric polynomial in the elements (equivalently in* * the nonzero elements) of the F2-vector space F2. We refer to [Wi ] fo* *r more detail. Note that what he denotes cn,iis what we call Dn-i(x1, . .,.xn). Lemma 4.6. For any n, Y D1(x1, . .,.xn+1)= (xn+1 + x) + D1(x1, . .,.xn)2 x2F2 Xn n 2n-i 2 = x2n+1+ xn+1Di(x1, . .,.xn) + D1(x1, . .,.xn) . i=1 Proof.The first equality is shown in [Wi , Proposition 1.3(b)]; here we prove t* *hem both simultaneously. Set Vn = F2. Since oei(Vn) = 0 whenever 2n - i i* *s not a power of 2 (cf. [Wi , Proposition 1.1]), X2n D1(x1, . .,.xn+1)= oei(Vn).oe2n-i(xn+1 + Vn) i=0 Y Xn = (xn+1 + x) + Di(x1, . .,.xn).oe2n-i(xn+1 + Vn). x2Vn i=1 Also, since oei(Vn) = 0 for 0 < i < 2n-1 as noted above, Xk ( n-1 n-i 0 if 0 < k < 2 oek(xn+1 + Vn) = xk-in+1.2k-ioei(Vn) = n-1 i=0 D1(x1, . .,.xn)if k = 2 . Ran Levi and Bob Oliver * * 33 This proves the first equality, and the second follows since Y n 2nX n n Xn n-i (xn+1 + x) = x2n+1+ x2n-i+1oei(Vn) = x2n+1+ x2n+1Di(x1, . .,.xn). x2Vn i=1 i=1 In the following proposition (and throughout the rest of the section), we wor* *k with the polynomial ring F2[x, y, z, w], with the natural action of GL4(F2). Let GL22(F2), GL31(F2) GL4(F2) be the subgroups of automorphisms of V def=F2which leave invariant * *the sub- spaces and , respectively. Also, let GL220(F2) GL22(F2) be th* *e subgroup of automorphisms which are the identity modulo . Thus, when described in * *terms of block matrices (with respect to the given basis {x, y, z, w}), 2 2 GL31(F2) = A0X1 , GL2(F2) = B0YC , and GL20(F2) = B0YI , for A 2 GL3(F2), X a column vector, B, C 2 GL2(F2), and Y 2 M2(F2). We need to make more precise the relation between V (or the polynomial ring F2[x, y, z, w]) and the cohomology of Spin(7). To do this, let W Spin(7) be* * the inverse image of the elementary abelian subgroup * * ff diag(-1, -1, -1, -1, 1, 1, 1), diag(-1, -1, 1, 1, -1, -1, 1), diag(-1, 1, -1* *, 1, -1, 1, -1) SO(7). Thus, W ~=C42. Fix a basis {j, j0, ,, i} for W , where i 2 Z(Spin(7)) is the no* *ntrivial element. Identify V = W *in such a way that {x, y, z, w} V is the dual bas* *is to {j, j0, ,, i}. This gives an identification H*(BW ; F2) = F2[x, y, z, w], arranged such that the action of NSpin(7)(W )=W on V = consists o* *f all automorphisms which leave invariant, and thus can be identified with * *the action of GL31(F2). Finally, set __ H = CSpin(7)(,) ~=Spin(4) xC2 Spin(3) ~=SU(2)3={ (I, I, I)} (the central product). Then in the same way, the action of N_H(W )=W on H*(BW ;* * F2) can be identified with that of GL220(F2). Proposition 4.7. The inclusions __ BW -----! BH -----! BSpin(7) -----! BDI(4) as defined above, together with the identification H*(BW ; F2) = F2[x, y, z, w]* *, induce isomorphisms A def=H*(BDI(4); F2) = F2[x, y, z, w]GL4(F2)= F2[a8, a12, a14, a15] 3(F ) B def=H*(BSpin(7); F2) = F2[x, y, z, w]GL1 2= F2[b4, b6, b7, b8](*) __ 2 C def=H*(BH ; F2) = F2[x, y, z, w]GL20(F2)= F2[c2, c3, c04, c004] ; where a8 = D1(x, y, z, w), a12= D2(x, y, z, w), a14= D3(x, y, z, w), a15= D4(x, y, * *z, w); Y b4 = D1(x, y, z), b6 = D2(x, y, z), b7 = D3(x, y, z), b8 = (w +* * ff); ff2 34 Construction of 2-local finite groups and Y Y c2 = D1(x, y), c3 = D2(x, y), c04= (z + ff), c004= (w + ff* *). ff2 ff2 Furthermore, __ (a)the natural action of 3 on H ~=SU(2)3={ (I, I, I)} induces the action on C * *which fixes c2, c3 and permutes {c04, c004, c04+ c004}; and (b)the above variables satisfy the relations a8= b8 + b24 a12= b8b4 + b26 a14= b8b6 + b27 a15= b8b7 b4= c04+ c22 b6= c2c04+ c23 b7= c3c04 b8= c004(c04+ c004) . Proof.The formulas for A = H*(BDI(4); F2) are shown in [DW1 ]. From [DW1 , Le* *m- mas 3.10 & 3.11], we see there are (some) identifications 3(F ) * __ GL2* *(F ) H*(BSpin(7); F2) ~=F2[x, y, z, w]GL1 2 and H (BH ; F2) ~=F2[x, y, z, w] 2* *0 2. __ From the explicit choices of subgroups W H Spin(7) as described above (and by the descriptions in Proposition_A.8 of the automorphism groups), the images * *of H*(BSpin(7); F2) and H*(BH ; F2) in F2[x, y, z, w] are seen to be contained in * *the rings of invariants, and hence these isomorphisms actually are equalities as cl* *aimed. We next prove the equalities in (*) between the given rings of invariants and* * poly- nomial algebras. The following argument was shown to us by Larry Smith. If k * *is a field and V is an n-dimensional vector space over k, then a system of parame* *ters in the polynomial algebra k[V ] is a set of n homogeneous elements f1, . .,.fn * *such that k[V ]=(f1, . .,.fn) is finite dimensional over k. By [Sm2 , Proposition 5.* *5.5], if V is an n-dimensional k[G]-representation, and f1, . .,.fn 2 k[V ]G is a system of p* *arame- ters the product of whose degrees is equal to |G|, then k[V ]G is a polynomial * *algebra with f1, . .,.fn as generators. By [Sm2 , Proposition 8.1.7], F2[x, y, z, w] is* * a free finitely generated module over the ring generated by its Dickson invariants (this holds * *for poly- nomial algebras over any Fp), and thus F2[x, y, z, w]=(a8, a12, a14, a15) is fi* *nite. (This can also be shown directly using the relation in Lemma 4.6.) So assuming the re* *lations in point (b), the quotients F2[x, y, z, w]=(b4, b6, b7, b8) and F2[x, y, z, w]=* *(c2, c3, c04, c004) are also finite. In each case, the product of the degrees of the generators is clea* *rly equal to the order of the group in question, and this finishes the proof of the last * *equality in the second and third lines of (*). It remains to prove points (a) and (b). Using Lemma 4.6, the ci are expresse* *d as polynomials in x, y, z, w as follows: c2 = D1(x, y) = x2 + xy + y2 c3 = D2(x, y) = xy(x + y) (1) c04= D1(x, y, z) + D1(x, y)2 = z4 + z2D1(x, y) + zD2(x, y) = z4 + z2c2 + zc3 c004= D1(x, y, w) + D1(x, y)2 = w4 + w2D1(x, y) + wD2(x, y) = w4 + w2c2 + wc3. In particular, Y c04+ c004= (z + w)4 + (z + w)2D1(x, y) + (z + w)D2(x, y) = (z + w +(ff).* *2) ff2 Ran Levi and Bob Oliver * * 35 Furthermore, by (1), we get Sq1(c2)= c3 Sq1(c3)= Sq1(c04) = Sq1(c004) = 0 Sq2(c3)= x2y2(x + y) + xy(x + y)3 = c2c3 Sq2(c04)= z4c2 + z2c22+ zc2c3 = c2c04 (3) Sq3(c04)= Sq1(c2c04) = c3c04 Sq2(c004)= c2c004 Sq3(c004)= c3c004. __ The permutation action_of_ 3 on H ~=SU(2)3={ (I, I, I)} permutes the three e* *l- ements i, ,, i + , of Z(H ) W , and thus (via the identification V = W *descr* *ibed above) induces the identity on x, y 2 V and permutes the elements {z, w, z + w}* * mod- 2(F ) ulo . Hence the induced action of 3 on C = F2[V ]GL20 2 is the restricti* *on of the action on F2[V ] = F2[x, y, z, w] which fixes x, y and permutes {z, w, z + w}. * *So by (1) and (2), we see that this action fixes c2, c3 and permutes the set {c04, c004, * *c04+ c004}. This proves (a). It remains to prove the formulas in (b). From (1) and (3) we get b4= D1(x, y, z) = c04+ c22, b6= D2(x, y, z) = Sq2(b4) = c2c04+ c23, b7= D3(x, y, z) = Sq1(b6) = c3c04. Also, by (1) and (2), Y i Y j i Y j b8 = (w + ff) = (w + ff) . (w + z + ff) = c004(c04+ c004* *). ff2 ff2 ff2 This proves the formulas for the bi in terms of ci. Finally, we have a8 = D1(x, y, z, w) = b8 + b24, a12= D2(x, y, z, w) = Sq4(b8 + b24) = Sq4(c004(c04+ c004) + (c04+ c22)2) = c04c004(c04+ c004) + c22c004(c04+ c004) + c22c042+ c43= b8b4 + b26 a14= D3(x, y, z, w) = Sq2(a12) = c2c04c004(c04+ c004) + c23c004(c04+ c00* *4) + c23c042 = b8b6 + b27 a15= D4(x, y, z, w) = Sq1(a14) = c3c04c004(c04+ c004) = b8b7; and this finishes the proof of the proposition. Lemma 4.8. Let ~ 2 Aut(C) be the algebra involution which exchanges c04and c004* *and leaves c2 and c3 fixed. An element of C will be called "~-invariant" if it is f* *ixed by this involution. Then the following hold: (a)If fi 2 B is ~-invariant, then fi 2 A. (b)If fi 2 B is such that fi.c04iis ~-invariant, then fi = fi0.bi8for some fi02* * A. Proof.Point (a) follows from Proposition 4.7 upon regarding A, B, and C as the * *fixed subrings of the groups GL4(F2), GL31(F2) and GL220(F2) acting on F2[x, y, z, w]* *, but also follows from the following direct argument. Let m be the degree of fi as a poly* *nomial 36 Construction of 2-local finite groups in b8; we argue by induction on m. Write fi = fi0 + bm8.fi1, where fi1 2 F2[b4* *, b6, b7], and where fi0 has degree < m (as a polynomial in b8). If m = 0, then fi = fi1 2 F2[b4, b6, b7] F2[c2, c3, c04], and hence fi 2 F2[c2, c3] since it is ~-invar* *iant. But from the formulas in Proposition 4.7(b), we see that F2[b4, b6, b7] \ F2[c2, c3] con* *tains only constant polynomials (hence it is contained in A). Now assume m 1. Then, expressed as a polynomial in c2, c3, c04, c004, the * *largest power of c004which occurs in fi is c0042m. Since fi is ~-invariant, the highest* * power of c04 which occurs is c042m; and hence by Proposition 4.7(b), the total degree of eac* *h term in fi1 (its degree as a polynomial in b4, b6, b7) is at most m. So for each term b* *r4bs6bt7in fi1, br4bs6bt7bm8- am-r-s-t8ar12as14at15 is a sum of terms which have degree < m in b8, and thus lies in A by the induct* *ion hypothesis. To prove (b), note first that since fi.c04iis ~-invariant and divisible by c0* *4i, it must also be divisible by c004i, and hence c004i|fi. Furthermore, by Proposition 4.7, all* * elements of B as well as c04are invariant under the involution which fixes c04and sends c0047* *! c04+ c004. Thus (c04+ c004)i|fi. Since b8 = c004(c04+ c004), we can now write fi = fi0.bi8* *for some fi 2 B. Finally, since fi.c04i= fi0.c04i.c004i.(c04+ c004)i is ~-invariant, fi0 is also ~-invariant, and hence fi02 A by (a). 2(F ) Note that C3 3 = GL2(F2) act on C = F2[x, y, z, w]GL20 2: via the action * *of GL22(F2)=GL220(F2), or equivalently by permuting c04, c004, and c04+ c004(and f* *ixing c2, c3). Thus A = B\CC3, since GL4(F2) is generated by the subgroups GL31(F2) and GL22(F* *2). This is also shown directly in the following lemma. Proposition 4.9. The following square is a pullback square, where all maps are * *induced by inclusions between the subrings of F2[x, y, z, w]: * C3 *A=F2__! C=F2 | | | | # # *B=F2___! *C=F2. Proof.Let ~ be the involution of Lemma 4.8: the algebra involution of C which e* *x- changes c04and c004and leaves c2 and c3 fixed. By construction, all elements in* * the image of *B=F2are invariant under the involution which fixes c04(and c2, c3), and se* *nds c004to c04+ c004. Hence elements in the image of *B=F2are fixed by C3 if and only if * *they are fixed by 3, if and only if they are ~-invariant. So it will suffice to show th* *at all of the above maps are injective, and that all ~-invariant elements in the image of *B* *=F2lie in the image of *A=F2. The injectivity is clear, and the square is a pullback for* * 0-=F2by Lemma 4.8. Fix a ~-invariant element ! = P1db4 + P2db6 + P3db7 + P4db8 (1) = P2c04dc2 + P3c04dc3 + P4c04dc004+ (P1 + P2c2 + P3c3 + P4c004) dc042 1* *B=F2, where Pi2 B for each i. By applying ~ to (1) and comparing the coefficients of * *dc2 and dc3, we see that P2c04and P3c04are ~-invariant. Also, upon comparing the coeffi* *cients of dc04, we get the equation P1 + P2c2 + P3c3 + P4c004= ~(P4)c004. (2) Ran Levi and Bob Oliver * * 37 So by Lemma 4.8, P2 = P20b8 and P3 = P30b8 for some P20, P302 A. Upon subtracti* *ng P20da14+ P30da15= P2db6 + P3db7 + (P20b6 + P30b7) db8 from ! and introducing an appropriate modification to P4, we can now assume that P2 = P3 = 0. With this assumption and (2), we have P1 + P4c004= ~(P4c04) = ~(P4).c004, so that P1c04= (P4 + ~(P4))c04c004 (3) is ~-invariant. This now shows that P1 = P10b8 for some P102 A, and upon subtra* *cting P10da12 from ! we can assume that P1 = 0. This leaves ! = P4db8 = P4da8. By (3) again, P4 is ~-invariant, so P4 2 A by Lemma 4.8 again, and thus ! 2 1A=F2. The remaining cases are proved using the same techniques, and so we sketch th* *em more briefly. To prove the result in degree two, fix a ~-invariant element ! = P1db4db6 + P2db4db7 + P3db4db8 + P4db6db7 + P5db6db8 + P6db7db8 = P4c042dc2dc3 + (P1c04+ P4c3c04+ P5c04c004) dc2dc04+ P5c042dc2dc004 + (P2c04+ P4c2c04+ P6c04c004) dc3dc04+ P6c042dc3dc004 + (P3c04+ P5c2c04+ P6c3c04) dc04dc0042 2B=F2. Using Lemma 4.8, we see that P4 = P40b28, and hence can assume that P4 = 0. One then eliminates P1 and P2, then P5 and P6, and finally P3. If ! = P1db4db6db7 + P2db4db6db8 + P3db4db7db8 + P4db6db7db8 = (P1c042+ P4c042c004) dc2dc3dc04+ (P2c042+ P4c3c042) dc2dc04dc004 + (P3c042+ P4c2c042) dc3dc04dc004+ P4c043dc2dc3dc0042 3B=F2 is ~-invariant, then we eliminate successively P1, then P4, then P2 and P3. Finally, if ! = P db4db6db7db8 = P c043dc2dc3dc04dc0042 4B=F2 is ~-invariant, then P = P 0b38for some P 02 A by Lemma 4.8 again, and so ! = P 0da8da12da14da152 4A=F2. Appendix Appendix A. Spinor groups over finite fields Let F be any field of characteristic 6= 2. Let V be a vector space over F ,* * and let b: V --! F be a nonsingular quadratic form. As usual, O(V, b) denotes the grou* *p of isometries of (V, b), and SO(V, b) the subgroup of isometries of determinant 1.* * We will be particularly interested in elementary abelian 2-subgroups of such ortho* *gonal groups. Lemma A.1. Fix an elementary abelian 2-subgroup E O(V, b). For each irreducib* *le character Ø 2 Hom (E, { 1}), let Vffl V denote the corresponding eigenspace:* * the subspace of elements v 2 V such that g(v) = Ø(g).v for all g 2 E. Then the rest* *riction of b to each subspace Vfflis nonsingular, and V is the orthogonal direct sum of* * the Vffl. 38 Construction of 2-local finite groups Proof.Elementary. We give a very brief sketch of the definition of spinor groups via Clifford a* *lgebras; for more details we refer to [Di, xII.7] or [As2 , x22]. Let T (V ) denote the * *tensor algebra of V , and set C(V, b) = T (V )=<(v v) - b(v)> : the Clifford algebra of (V, b). To simplify the notation, we regard F as a sub* *ring of C(V, b), and V as a subgroup of its additive group; thus the class of v1 . .* * .vk will be written v1. .v.k. Note that vw + wv = 0 if v, w 2 V and v ? w. Hence if dim F(V ) = n, and {v1, . .,.vn} is an orthogonal basis, then the set of 1 a* *nd all vi1. .v.ikfor i1 < . .<.ik (1 k n) is an F -basis for C(V, b). Write C(V, b) = C0 C1, where C0 and C1 consist of classes of elements of ev* *en or odd degree, respectively. Let G C(V, b)* denote the group of invertible eleme* *nts u such that uV u-1 = V , and let ß :G --! O(V, b) be the homomorphism ( (v 7! -uvu-1) if u 2 C1 ß(u) = (v 7! uvu-1) if u 2 C0. In particular, for any nonisotropic element v 2 V (i.e., b(v) 6= 0), v 2 G and * *ß(v) is the reflection in the hyperplane v? . By [Di, xII.7], ß is surjective and Ker(ß* *) = F *. Let J be the antiautomorphism of C(V, b) induced by the antiautomorphism v1 . . .vk 7! vk . . .v1 of T (V ). Since O(V, b) is generated by hyperplane ref* *lections, G is generated by F *and nonisotropic elements v 2 V . In particular, for any * *u = ~.v1. .v.k2 G, J(u).u = ~2 . vk. .v.1. v1. .v.k= ~2 . b(v1) . .b.(vk) 2 F *= Ker(ß); implying that ß(J(u)) = ß(u)-1 for all u 2 G. There is thus a homomorphism e`:G -------! F * defined by e`(u) = u.J(u). In particular, e`(~) = ~2 for ~ 2 F * G, while for any set of nonisotropic ele* *ments v1, . .,.vk of V , e`(v1. .v.k) = (v1. .v.k)(vk. .v.1) = b(v1) . .b.(vk). Hence e`factors through a homomorphism `V,b:O(V, b) -------! F *=F *2= F *={u2| u 2 F *}, called the spinor norm. Set G+ = ß-1(SO(V, b)) = G \ C0, and define Spin(V, b) = Ker(e`|G+ ) and (V, b) = Ker(`V,b|SO(V,b)). In particular, (V, b) has index 2 in SO(V, b) if F is a finite field, and (V,* * b) = SO(V, b) if F is algebraically closed (all units are squares). We thus get a co* *mmutative diagram Ran Levi and Bob Oliver * * 39 1 1 1 fflffl| fflffl||~7!~2 fflffl|| 1 ____//_{ 1}__________//F_*_________//F *2__//1 | || || | | | | | | fflffl| fflffl|e` fflffl| (A.2) 1 __//_Spin(V,_b)______//G+__________//_F_*__//_1 | | | | | | i | | | | | fflffl| fflffl|`V,b fflffl| 1 ____// (V, b)_____//SO(V, b)_____//F *=F *2//_1 fflffl| fflffl| fflffl| 1 1 1 where all rows and columns are short exact, and where all columns are central e* *xten- sions of groups. If dim(V ) 3 (or if dim(V ) = 2 and the form b is hyperbolic* *), then (V, b) is the commutator subgroup of SO(V, b) [Di, xII.8]. The following lemma follows immediately from this description of Spin(V, b), * *together with the analogous description of the corresponding spinor group over the algeb* *raic closure of F . __ __ __ _ Lemma A.3. Let F be the algebraic closure of F , and set V = F F V and b= Id_F* * b. __ _ Then Spin(V, b) is the subgroup_of Spin(V , b) consisting of those elements fix* *ed by all Galois automorphisms _ 2 Gal(F =F ). For any nonsingular quadratic form b on a vector space V , the discriminant o* *f b (or of V ) is the determinant of the corresponding symmetric bilinear form B, relat* *ed to b by the formulas b(v) = B(v, v) and B(v, w) = 1_2b(v + w) - b(v) - b(w) . Note that the discriminant is well defined only modulo squares in F *. When W * * V is a subspace, then we define the discriminant of W to mean the discriminant of* * b|W . In what follows, we say that the discriminant of a quadratic form is a square o* *r a nonsquare to mean that it is the identity or not in the quotient group F *=F *2. Lemma A.4. Fix an involution x 2 SO(V, b), and let V = V+ V- be its eigenspace decomposition. Then the following hold. (a)x 2 (V, b) if and only if the discriminant of V- is a square. (b)If x 2 (V, b), then it lifts to an element of order 2 in Spin(V, b) if and * *only if dim (V-) 2 4Z. (c)If x 2 (V, b), and if ff 2 (V, b) is such that [x, ff] = 1, then ff = ff+ * * ff-, where ff 2 O(V , b). Also, the liftings of x and ff commute in Spin(V, b) if and * *only if det(ff-) = 1. Proof.Let {v1, . .,.vk} be an orthogonal basis for V- (k is even). Then x = ß(v* *1. .v.k) in the above notation, since ß(vi) is the reflection in the hyperplane vi?. Hen* *ce by the commutativity of Diagram (A.2), `V,b(x) b(v1). .b.(vk) = det(b|V-) (mod F *2). Thus x 2 (V, b) = Ker(`V,b) if and only if V- has square discriminant. 40 Construction of 2-local finite groups In particular, if x 2 (V, b), then the product of the b(vi) is a square, and* * hence (upon replacing v1 by a scalar multiple) we can assume that b(v1). .b.(vk) = 1.* * Then exdef=v1. .v.k2 Spin(V, b) = Ker(e`). Since vw = -wv in the Clifford algebra wh* *enever v ? w, and since (vi)2 = b(vi) for each i, ( 1 if k 0 (mod 4) ex2= (-1)k(k-1)=2.(v1)2. .(.vk)2 = (-1)k(k-1)=2= -1 if k 2 (mod 4) . This proves (b). It remains to prove (c). The first statement (ff = ff+ ff-) is clear. Fix* * liftings eff2 C(V , b)*. Rather than defining the direct sum of an element of C(V+, b) w* *ith an element of C(V-, b), we regard the groups C(V , b)* as (commuting) subgroups* * of C(V, b)*, and set eff= eff+Oeff-= eff-Oeff+2 Spin(V, b). Let ex= v1. .v.kbe as above. Clearly, excommutes with all elements of C(V+, b).* * Since (v1. .v.k).vi= (-1)k-1.vi.(v1. .v.k) = -vi.(v1. .v.k) for i = 1, . .,.k, we have ex.fi = (-1)i.fi.exfor all fi 2 Ci(V-, b) (i = 0, 1)* *. In particular, since [eff+, eff-] = 1, [ex, eff] = [ex, eff-] = det(ff-), and this finishes th* *e proof. We will need explicit isomorphisms which describe Spin3(F ) and Spin4(F ) in * *terms of SL2(F ). These are constructed in the following proposition, where M02(F ) d* *enotes the vector space of matrices of trace zero. Note that the determinant is a nons* *ingular quadratic form on M2(F ) and on M02(F ), in both cases with square discriminant. Proposition A.5. Define æ3: SL2(F ) ------! (M02(F ), det) and æ4: SL2(F ) x SL2(F ) ------! (M2(F ), det) by setting æ3(A)(X) = AXA-1 and æ4(A, B)(X) = AXB-1. Then æ3 and æ4 are both epimorphisms, and lift to unique isomorphisms je3 0 SL2(F ) ------!~ Spin(M2(F ), det) = and je4 SL2(F ) x SL2(F ) ------!~ Spin(M2(F ), det). = Proof.See [Ta , pp. 142, 199] for other ways of defining these isomorphisms. By* * Lemma A.3, it suffices to prove this (except for the uniqueness of the lifting) when * *F is alge- braically closed. In particular, (M02(F ), det) = SO(M02(F ), det) and (M2(F * *), det) = SO(M2(F ), det) in this case. For general V and b, the group SO(V, b) is generated by reflections fixing n* *on- isotropic subspaces (i.e., of nonvanishing discriminant) of codimension 2 (cf. * * [Di, xII.6(1)]). Hence to see that æ3 and æ4 are surjective, it suffices to show th* *at such elements lie in their images. A codimension 2 reflection in SO(M02(F ), det) is* * of the form RX (the reflection fixing the line generated by X) for some X 2 M02(F ) wh* *ich is nonisotropic (i.e., det(X) 6= 0). Since F is algebraically closed, we can a* *ssume X 2 SL2(F ). Then X2 = -I (since Tr(X) = 0 and det(X) = 1), and RX = æ3(X) since it has order 2 and fixes X. Thus æ3 is onto. Ran Levi and Bob Oliver * * 41 Similarly, any 2-dimensional nonisotropic subspace W V has an orthonormal b* *asis {Y, Z}, and ZY -1and Y -1Z have trace zero (since they are orthogonal to the id* *entity matrix) and determinant one. Hence their square is -I, and one repeats the abo* *ve argument to show that RW = æ4(ZY -1, Y -1Z). So æ4 is onto. The liftings eæmexist and are unique since SL2(F ) is the universal central e* *xtension of P SL2(F ) (or universal among central extensions by 2-groups if F = F3). We now restrict to the case F = Fq where q is an odd prime power. We refer to [As2 , x21] for a description of quadratic forms in this situation, and the not* *ation for the associated orthogonal groups. If n is odd and b is any nonsingular quadrati* *c form on Fnq, then every nonsingular quadratic form is isomorphic to ub for some u 2 * *F*q, and hence one can write SOn(q) = SO(Fnq, b) = SO(Fnq, ub) without ambiguity. If n is even, then there are exactly two isomorphism classes of quadratic forms on* * Fnq; and one writes SO+n(q) = SO(Fnq, b) when b is the hyperbolic form (equivalently, has discriminant (-1)n=2 modulo squares), and SO-n(q) = SO(Fnq, b) when b is not hyperbolic (equivalently, has discriminant (-1)n=2.u for u 2 F*qnot a square). * * This notation extends in the obvious way to n(q) and Spinn(q). The following lemma does, in fact, hold for for orthogonal representations ov* *er arbi- trary fields of characteristic 6= 2. But to simplify the proof (and since we we* *re unable to find a reference), we state it only in the case of finite fields. Lemma A.6. Assume F = Fq, where q is a power of an odd prime. Let V be an F -vector space, and let b be a nonsingular quadratic form on V . Let P O(V, * *b) be a 2-subgroup which is orthogonally irreducible; i.e., such that V has no splitt* *ing as an orthogonal direct sum of nonzero P -invariant subspaces. Then dimF (V ) is a po* *wer of 2; and if dim(V ) > 1 then b has square discriminant. Proof.This means showing that each orthogonal group O(Fqn, b), such that either* * n is not a power of 2, or n = 2k 2 and the quadratic form b has nonsquare discrimi* *nant, contains some subgroup Om (q) x On-m (q) (for 0 < m < n) of odd index. We refer* * to the standard formulas for the orders of these groups (see [Ta , p.165]): if ffl* * = 1 then n-1Y Yn 2 * *2i |Offl2n(q)| = 2qn(n-1)(qn - ffl) (q2i- 1) and |O2n+1(q)| = 2qn (q* * - 1). i=1 i=1 We will also use repeatedly the fact that for all 0 < i < 2k (k 1), the large* *st powers k+i i 2k+i * * i of 2 dividing (q2 - 1) and (q - 1) are the same. In other words, (q - 1)=(* *q - 1) is invertible in Z(2). For any n 1, __|O2n+1(q)|__ n qn + ffl = q .______ |Offl2n(q)|.|O1(q)| 2 is odd for an appropriate choice of ffl. Thus, there are no irreducibles of odd* * dimension. Assume n is not a power of 2, and write n = 2k + m where 0 < m < 2k and k 1. Then _m-1 k ! _ k ! _ k ! _____|Offl2n(q)|_ m2k+1 Y q2(2 +i)- 1 q2 +m - ffl q2 + 1 = q . __________ . _________ . _______ , |O+2k+1(q)|.|Offl2m(q)| i=1 q2i- 1 qm - ffl 2 42 Construction of 2-local finite groups and each factor is invertible in Z(2). When n = 2m = 2k and k 1, then O-2n(q)* * is the orthogonal group for the quadratic form with nonsquare discriminant, and _ m-1 ! ____|O-2n(q)|___ 2m2 Y q2(m+i)- 1 q2m + 1 = q . __________ ._______, |O+2m(q)|.|O-2m(q)| i=1 q2i- 1 2 and again each factor is invertible in Z(2). Finally, ___|Offl2(q)|_ q - ffl = _____ |O1(q)|.|O1(q)| 2 is odd whenever q 1 (mod 4) and ffl = -1, or q 3 (mod 4) and ffl = +1; and * *these are precisely the cases where the quadratic form on Fq2 has nonsquare discriminant. We must classify the conjugacy classes of those elementary abelian 2-subgroup* *s of Spin7(q) which contain its center. The following definition will be useful when* * doing this. Definition A.7. Fix an odd prime power q, and identify SO7(q) = SO(F7q, b) and Spin7(q) = Spin(F7q, b), where b is a nonsingular quadratic form with square di* *scrimi- nant. An elementary abelian 2-subgroup of SO7(q) or of Spin7(q) will be called * *of type I if its eigenspaces all have square discriminant (with respect to b), and of t* *ype II oth- erwise. Let En be the set of elementary abelian 2-subgroups in Spin7(q) which c* *ontain Z(Spin7(q)) ~=C2 and have rank n. Let EInand EIInbe the subsets of En consistin* *g of those subgroups of types I and II, respectively. In the following two propositions, we collect together the information which * *will be needed about elementary abelian 2-subgroups of Spin7(q). We fix Spin7(q) = Spin(V, b), where V ~=F7q, and b is a nonsingular quadratic form with square d* *is- criminant. Let z 2 Z(Spin7(q))_be_the generator. For any subgroup H Spin7(q) * *or _ any element g 2 Spin7(q), let H and gdenote their images in 7(q) SO7(q)._For* * each elementary abelian 2-subgroup E Spin7(q), and each character Ø 2 Hom (E , { 1* *}), Vffl V denotes the eigenspace of Ø (and V1 denotes the eigenspace of the trivi* *al char- acter). Also (when z 2 E), Aut (E, z) denotes the group of all automorphisms o* *f E which send z to itself. Proposition A.8. For any odd prime power q, the following table describes the n* *um- bers of Spin7(q)-conjugacy classes in each of the sets EInand EIIn, the dimensi* *ons and discriminants of the eigenspaces of subgroups in these sets, and indicates in w* *hich cases AutSpin7(q)(E) contains all automorphisms which fix z. ______________________________________________________________ | Set of subgroups |E|I | EI | EII | EI |EII | |___________________________|_|2__|__3___|__3___|___4___|_4__|__ | Nr. conj. classes |1| | 1 | 1 | 2 | 1 | |____________________________|_|__|______|______|______|______| | dim(V ) | |3 | 1 | 0 | |______1___________________|_|____|_____________|____________|_ | dim(V ), Ø 6= 1 |4| | 2 | 1 | |______ffl__________________|_|__|______________|____________|_ | discr(V , b) sq|u|ares|quaren|onsq. |_ | _ | |________1___________________|_|____|______|______|____|_____|_ | discr(V , b), Ø 6= 1 squ|a|res|quaren|onsq.s|quare |both | |________ffl__________________|_|___|______|______|______|____ | | Aut (E) = Aut(E, z) |y|es |yes |yes | yes |no | |_____Spin7(q)_____________|_|_____|______|______|_______|____| There are no subgroups in E2 of type II, and no subgroups of rank 5. Furtherm* *ore, we have: (a)For all E 2 E4, CSpin7(q)(E) = E. Ran Levi and Bob Oliver * * 43 __ __ (b)If E, E02 EI4, then E0 = gE g-1 for some g 2 SO7(q), and E and E0 are Spin7(* *q)- conjugate if and only if g 2 7(q). _ _ __ (c)If E 2 EII4, then there is_a unique element 1 6= x = x(E) 2 E with the prop* *erty _ that for 1 6= Ø 2 Hom (E , { 1}), Vfflhas square discriminant if Ø(x) = 1 a* *nd _ * * __ nonsquare discriminant if Ø(x) = -1. Also, the_image of AutSpin7(q)(E) in Au* *t(E ) is the group of all automorphisms which send x to itself; and if X E denot* *es _ __ * * __ the inverse image of E, then AutSpin7(q)(E) contains all automorphisms * *of E which are the identity on X and the identity modulo . (d)If E 2 E3, then CSpin7(q)(E) = Ao C2, where A is abelian and C2 acts on A by inversion. If E 2 EII3, then the Sylow 2-subgroups of CSpin7(q)(E) are elem* *entary abelian of rank 4 (and type II). Proof.Write Spin= Spin7(q) for short. Fix an elementary abelian subgroup E Sp* *in such that z 2 E. Step 1: We first_show that rk(E) 4, and that the dimensions of the eigenspa* *ces Vfflfor Ø 2 Hom (E , { 1}) are as described in the table. __ By Lemma A.4, every involution_in E has a 4-dimensional (-1)-eigenspace.__In particular, if rk(E) = 2, (E ~=C2), then dim (Vffl) = 4 for 1 6= Ø 2 Hom (E , * *{ 1}), while dim(V1) = 3. Now assume rk(E) = n for some n > 2. Assume we_have_shown, for all E0 2 En-1, that the_eigenspace of the trivial character of E 0is r-dimensional._For each 1* * 6= Ø 2 Hom (E , { 1}), let Effl2 En-1 be the subgroup_such that E ffl= Ker(Ø); then V1* * Vfflis the eigenspace of the trivial character of Effl= Ker(Ø), and thus dim(V1)+dim (* *Vffl) = r. Hence all nontrivial characters of E have_eigenspaces of the same dimension. Si* *nce there are 2n-1- 1 nontrivial characters of E, we have dim(V1) + (2n-1- 1) dim(Vffl) =* * 7, and these two equations completely determine dim(V1) and dim(Vffl). Using this proc* *edure, the dimensions of the eigenspaces are shown inductively to be equal_to those gi* *ven by the table. Also, this shows that rk(E) 4, since otherwise rk(E ) 4, so the * *Vfflfor Ø 6= 1 must be trivial (they cannot all have dimension 1), so E acts on V via* * the identity, which contradicts the assumption that E Spin7(q). Step 2: We next_show_that EII2= ;, describe the discriminants of the eigenspa* *ces of characters of E for E 2 En (for all n), and show that subgroups of rank 4 ar* *e self centralizing. In particular, this proves (a) together with the first statement * *of (c). If E 2 E2, then_E = for some noncentral involution g 2 Spin7(q), and t* *he _ eigenspaces of E = have square discriminant by Lemma A.4(a) (and since the ambient space V has square discriminant by assumption). Thus EII2= ;. __ _ If E 2 E3, then the sum of any two eigenspaces of E_is an eigenspace of g for* * some g 2 Er . Hence the sum of any two eigenspaces of E has square discriminant, * *so either all of the eigenspaces have square discriminant (E 2 EI3), or all of the* * eigenspaces have nonsquare discriminant (E 2 EII3). __ Assume rk(E) = 4. We have seen that all eigenspaces_of E are 1-dimensional. By Lemma A.4(c), for each a 2 CSpin7(q)(E), a(Vffl) =_Vfflfor each Ø 6= 1, and * *since dim(Vffl) = 1 it must act on each Vfflvia Id._Thus a 2 7(q) has order 2; let* * V be its eigenspaces. Then dim(V-) is even since det(a) = 1, and V- has square discrimin* *ant 44 Construction of 2-local finite groups by Lemma A.4(a). If dim(V-) = 4, then |a| = 2 (Lemma A.4(b)), and hence a 2 E since otherwise would have rank 5. If dim (V-)_= 2, then V- is the sum * *of the_eigenspaces_of two distinct_characters_Ø1,_Ø2 of E , there is some g 2 E su* *ch that Ø1(g) 6= Ø2(g), hence det(g|V-) = Ø1(g)Ø2(g) = -1, so [g, a] = z by Lemma A.4(c* *), and this contradicts the assumption that [a, E] = 1. If dim(V-) = 6, then_V- is* * the sum of the eigenspaces of all but one of the nontrivial characters of E , and t* *his gives a similar contradiction to the assumption [a, E] = 1. Thus, CSpin7(q)(E) = E. _ Now assume that E 2 EII4, and let x 2 O7(q) be the element which acts via - Id on eigenspaces with nonsquare discriminant, and via the identity on those with * *square_ discriminant. Since b has square discriminant on_V , the number of eigenspaces * *of E on which the discriminant is nonsquare is even, so x 2 7(q) by Lemma A.4(a),_and * *lifts to an element x 2 Spin7(q). Also, for each g 2 E, the (-1)-eigenspace of g has * *square discriminant_(Lemma A.4(a) again), hence contains an even number of eigenspaces* * of E of nonsquare discriminant, and by Lemma A.4(c) this shows that [g, x] = 1. Th* *us x 2 CSpin7(q)(E) = E, and this proves the first statement in (c). Step 3: We next check the numbers of conjugacy classes of subgroups in each of* * the sets EInand EIIn, and describe AutSpin(E) in each case. This finishes the proof* * of (b) and (c), and of all points in the above table. From the above description, we see immediately_that_if E and E0 have the same _ _ 0 rank and type, then any isomorphism ff 2 Iso(E , E0),_such_that ff(x(E)) = x(E * *) if E, E0 2 EII4, has the property that for all Ø 2 Hom (E 0, { 1}), Vffland VfflOf* *fhave the same dimension and the same discriminant (modulo squares). Hence for any such f* *f, __ __ there is an element g 2 O7(q) such that g(VfflOff) = Vfflfor all Ø; and ff = cg* * 2 Iso(E , E0) for such g. Upon replacing g by -g if necessary, we can assume that g 2 SO7(q).* * This shows that __ __ E, E0 have the same rank and type =) E and E 0are SO7(q)-conjugate (1) and also that ( __ __ Aut(E ) if E =2EII Aut SO7(q)(E ) = __ _ 4 (2) Aut(E , x(E))if E 2 EII4. We next claim that __ E =2EI4=) 9fl 2 SO7(q)r 7(q) such that[fl, E] = 1 . (3) To prove this, choose 1-dimensional nonisotropic_summands W Vffland W 0 V_, where Ø, _ are two distinct characters of E, and where W has square discriminan* *t and W 0has nonsquare discriminant._Let fl 2 SO7(q) be the involution_with (-1)-eige* *nspace W W 0. Then [fl, E] = 1, since fl sends each eigenspace of E to itself, and f* *l =2 7(q) since its (-1)-eigenspace has nonsquare discriminant (Lemma A.4(a)). __ If E has rank 4 and type I, then Aut (E ) ~= GL3(F2) is simple,_and in partic* *ular has no subgroup of index 2. Hence by (2), each element of Aut (E ) is_induced * *by conjugation by an_element_of 7(q). Also, if g 2 SO7(q) centralizes E, then g(V* *ffl) = Vffl for all Ø 2 Hom (E , { 1}), g acts via - Idon an even number of eigenspaces (si* *nce it has determinant +1), and hence g 2 7(q) by Lemma A.4(a). Thus __ E 2 EI4=) NSO7(q)(E ) 7(q) (4) Ran Levi and Bob Oliver * * 45 If E =2EI4, then by (3), for any g 2 SO7(q), there is fl 2 SO7(q)r__7(q)_such* * that__ __ cg|E = cgfl|E, and either g or gfl lies in 7(q). Thus IsoSO7(q)(E , E0) = Iso * *7(q)(E , E0) for any E0. Together with (1), this shows that E is Spin-conjugate to all other sub* *groups of the same rank and type, and together with (2) it shows that ( __ __ Aut(E ) if E =2EII4 Im Aut Spin(E) --! Aut(E ) = __ _ (5) Aut(E , x(E))if E 2 EII4. __ __ __ If E 2 EI4, then by (4) and (2), Aut 7(q)(E ) = Aut SO7(q)(E ) = Aut_(E ),_and* *_so (5) also holds in this case. Furthermore, for any g 2 SO7(q)r 7(q), E and gE g-1* * are representatives for two distinct_ 7(q)-conjugacy classes _ since by (4), no ele* *ment of the coset g. 7(q) normalizes E . We have now determined in all cases the number of conjugacy_classes_of subgro* *ups of a given rank and type, and the image of AutSpin(E) in Aut(E ). We next claim* * that if rk(E) < 4 or E 2 EI4, then fi E =2EII4=) Aut Spin(E) ff 2 Aut(E) fiff(z) = z, ff Id(mod ) .(6) Together with (5), this will finish the proof that AutSpin(E) is the group of a* *ll auto- morphisms of E which send z to itself. We also claim that fi E 2 EII4=) Aut Spin(E) ff 2 Aut(E) fiff|X = IdX, ff Id(mod ) ,(7) _ __ where X E denotes the inverse image of E , and this will finish the * *proof of (c). We prove (6) and (7) together._ Fix ff 2 Aut(E) (ff 6= Id) which sends z to i* *tself, induces the identity_on E , and such that ff|X = IdX if E 2 EII4. Then ther* *e is _ 1 6=_Ø 2 Hom (E , { 1}) such that ff(g) = g when Ø(g) = 1 and ff(g) = zg when Ø(g) = -1. Choose any character _ such that V_ 6= 0 and V_ffl6= 0, and let W * *V_ and W 0 V_fflbe 1-dimensional nonisotropic_subspaces with_the same discriminant (this is possible when E 2 EII4since x(E)_2 Ker(Ø)). Let g 2 O7(q) be the_invol* *ution whose (-1)-eigenspace is W W 0. Then g 2 7(q) by Lemma A.4(a), so g lifts to g 2 Spin7(q), and using Lemma A.4(c) one sees that cg = ff. Step 4: It remains_to prove (d). Assume E 2 E3. Let 1 = Ø1, Ø2, Ø3, Ø4 be the * *four characters of E , and set Vi = Vffli. Then dim (V1) = 1, dim (Vi) = 2 for i = * *2, 3, 4, and the Vi either all have square_discriminantLor all have nonsquare discrimina* *nt. For each g 2 CSpin(E), we can write g = 4i=1gi, where gi 2 O(Vi, bi). For each pa* *ir of distinct indices i, j 2 {2, 3, 4}, there is some g 2 E whose (-1)-eigenspace is* * Vi Vj, and hence det(gi gj) = 1 by Lemma A.4(c). This shows that the giall have the s* *ame determinant. Let A CSpin(E) be the subgroup of index 2 consisting of those g * *such that det(gi) = 1 for all i. _ _ _ Now, SO1(F q) = 1, while SO2(F q) ~=F*qis the group of diagonal matrices of t* *he form _ diag(u, u-1) with respect_to a hyperbolic basis of Fq2. Thus A is contained_in * *a central extension of C2 by (F *q)3, and any such extension is abelian since H2((F *q)3)* * = 0. Hence A is abelian. The groups_O2 (q) are all dihedral (see [Ta , Theorem 11.4]). Hen* *ce for any g 2 CSpin(E)r A, ghas order 2 and (-1)-eigenspace of dimension 4 (its inter* *section with each Viis 1-dimensional), and hence |g| = 2 by Lemma A.4(b). Thus all elem* *ents of CSpin(E)r A have order 2, so the centralizer must be a semidirect product of* * A with a group of order 2 which acts on it by inversion. 46 Construction of 2-local finite groups Now assume that E 2 EII3; i.e., that the Vi all have nonsquare discriminant. * *Then for i = 2, 3, 4, SO(Vi, bi) has order q 1, whichever is not a multiple of 4 (* *see [Ta , Theorem 11.4] again). Thus if g 2 A CSpin(E) has 2-power order, then gi = I* * _ for each i, the number of i for which gi = Id is even (since the (-1)-eigenspac* *e of g has square discriminant), and hence g 2 E. In other words, E 2 Syl2(A). A Syl* *ow 2-subgroup of CSpin(E) is thus generated by E together with an element of order* * 2 which acts on E by inversion; this is an elementary abelian subgroup of rank 4,* * and is necessarily of type II. We also need some more precise information_about the subgroups of Spin7(q) of rank 4 and type II. Let _q 2 Aut(Spin7(F q)) denote the automorphism induced by* * the field automorphism (x 7! xq). By Lemma A.3, Spin7(q) is precisely the subgroup* * of elements fixed by _q. Proposition A.9. Fix an odd prime power q, and let z 2 Z(Spin7(q)) be the centr* *al involution. Let C and C0 denote the two conjugacy classes of subgroups E Spin* *7(q) of rank 4 and type I. Then the following hold. _ (a)For each E 2 E4, there is an element a 2 Spin7(F q) such that aEa-1 2 C. For* * any such a, if we set xC(E) def=a-1_q(a), then xC(E) 2 E and is independent of the choice of a. (b)E 2 C if and only if xC(E) = 1, and E 2 C0 if and only if xC(E) = z. (c)Assume E 2 EII4, and set ø(E) = . Then rk(ø(E)) = 2, and fi Aut Spin7(q)(E) = ff 2 Aut(E) fiff|fi(E)= Id . __ _____ The four eigenspaces of E contained in the (-1)-eigenspace of xC(E) all have* * non- square discriminant, and the other three eigenspaces all have square discrim* *inant. Proof.(a) For all E 2 E4, E has type I as a subgroup of Spin7(q2) since all el* *ements_ of Fq are squares in Fq2. Hence by Proposition A.8(b), for all E0 2 C, there i* *s a 2 ____-1 __0 _ 4 SO7(q2) 7(q4) such that_aE a = E . Upon lifting a to a 2 Spin7(q ), this pr* *oves that there is a 2 Spin7(F q) such that aEa-1 2 C. Fix any such a, and set x = xC(E) = a-1_q(a). For all g 2 E, _q(g) = g and _q(aga-1) = aga-1 since E, aEa-1 Spin7(q), and h* *ence aga-1 = _q(a).g._q(a-1) = a(xgx-1)a-1. Thus, x 2 CSpin7(_Fq)(E), and so x 2 E since it is self centralizing in each Sp* *in7(qk) (Proposition A.8(a)). * * _ We next check that xC(E) is independent of the choice of a. Assume a, b 2 Spi* *n7(F q) are such that aEa-1 2 C and bEb-1 2 C. Then by Proposition A.8(b), there is g 2 Spin7(q) such that gbE(gb)-1 = aEa-1. Set E0 = aEa-1 2 C, then gba-1 2 NSpin7(_Fq)(E0). Furthermore, since Aut Spin7(q)(E0) contains all automorphism* *s which send z to itself, and since E0 is self centralizing in each of the groups Spin7* *(qk) (both by Proposition A.8 again), we see that NSpin7(_Fq)(E0) is contained in Spin7(q)* *. Thus, ba-1 2 Spin7(q), so _q(ba-1) = ba-1; and this proves that xC(E) = a-1_q(a) = b-* *1_q(b) is independent of the choice of a. Ran Levi and Bob Oliver * * 47 (b) If E 2 C, then we can choose a = 1, and so xC(E) = 1. _ If E 2 C0, then by Proposition A.8(b), there is a 2 Spin7(q2) such that a* * 2 SO7(q)r 7(q)_and aEa-1_2 C._ Then _q(a) 6= a since a 2= Spin7(q) (Proposition A.3), but _q(a) = a since a 2 SO7(q). Thus, xC(E) = a-1_q(a) = z in this case. We have now shown that_xC(E) 2 if E has type I, and it remains to prove t* *he converse._Fix_a 2 Spin7(F_q) such that aEa-1 2 C. If xC(E) 2 , then _q(a) 2 * *{a, za}, so _q(a) = a, and hence a 2 SO7(q). Conjugation by an element of SO7(q) sends eigenspaces with square discriminant to eigenspaces with square discriminant, s* *o all eigenspaces of E must have square discriminant since all eigenspaces of aEa-1 d* *o. Hence E has type I. (c) Now write Spin = Spin7(q) for short. Assume E 2 EII4, and set x = xC(E) and ø(E) = . Then x =2 by (b), and thus ø(E) has rank 2. By (a) (the uniqueness of x having the given properties), each element of Aut* *Spin(E) restricts to the identity on ø(E). We have already seen (Proposition A.8(c)) th* *at there _ __ __ is an element x(E) 2 E such that the image in Aut (E ) of Aut Spin(E) is the g* *roup _ _ _ __ of automorphisms which fix x(E), and this shows that x(E) = x: the image in E of x. Since we already showed (Proposition A.8(c) again) that Aut Spin(E) contain* *s all automorphisms which are the identity on ø(E) and the identity modulo , this * *finishes the proof that AutSpin(E) is the group of all automorphisms which are the ident* *ity on ø(E). The last statement (about the discriminants of the eigenspaces) follows d* *irectly from the first statement of Proposition A.8(c). Throughout the rest of the section, we collect some more technical results wh* *ich will be needed in Sections 2 and 4. Lemma A.10. Fix k 2. Let A = e13(2k-1) 2 GL3(Z=2k) be the elementary matrix which has off diagonal entry 2k-1 in position (1, 3). Let T1 and T2 be the two * *maximal parabolic subgroups of GL3(2): T1 = GL12(Z=2) = (aij) 2 GL3(2) | a21= a31= 0 and T2 = GL21(Z=2) = (aij) 2 GL3(2) | a31= a32= 0 . Set T0 = T1 \ T2: the group of upper triangular matrices in GL3(2). Assume that ~i:Ti-----! SL3(Z=2k) are lifts of the inclusions (for i = 1, 2) such that ~1|T0 = ~2|T0. Then there* * is a homomorphism ~: GL3(2) ---! SL3(Z=2k) such that ~|T1= ~1, and either ~|T2= ~2, or ~|T2= cA O~2. Proof.We first claim that any two liftings oe, oe0:T2 --! SL3(Z=2k) are conjuga* *te by an element of SL3(Z=2k). This clearly holds when k = 1, and so we can assume inductively that oe oe0(mod 2k-1). Let M03(F2) be the group of 3 x 3 matrices* * of trace zero, and define æ: T2 --! M03(F2) via the formula oe0(B) = (I + 2k-1æ(B)).oe(B) for B 2 T2. Then æ is a 1-cocycle. Also, H1(T2; M03(F2)) = 0 by [DW1 , Lemma 4* *.3] (the module is F2[T2]-projective), so æ is the coboundary of some X 2 M03(F2), * *and oe and oe0differ by conjugation by I + 2k-1X. 48 Construction of 2-local finite groups By [DW1 , Theorem 4.1], there exists a section ~ defined on GL3(2) such that* * ~|T1= ~1. Let B 2 SL3(Z=2k) be such that ~|T2 = cB O~2. Since ~|T0 = ~2|T0, B must commute with all elements in ~(T0), and one easily checks that the only such el* *ements are A = e13(2k-1) and the identity. Recall that a p-subgroup P of a finite group G is p-radical if NG(P )=P is p-* *reduced; i.e., if Op(NG(P )=P ) = 1. (Here, Op(-) denotes the largest normal p-subgroup.* *) We say here that P is Fp(G)-radical if OutG (P ) (= OutFp(G)(P )) is p-reduced. In* * Section 4, some information will be needed involving the F2(Spin7(q))-radical subgroups* * of Spin7(q) which are also 2-centric. We first note the following general result. Lemma A.11. Fix a finite group G and a prime p. Then the following hold for any p-subgroup P G which is p-centric and Fp(G)-radical. (a)If G = G1x G2, then P = P1x P2, where Pi is p-centric in Gi and Fp(Gi)-radic* *al. (b)If P H C G, then P is p-centric in H and Fp(H)-radical. (c)If H C G has p-power index, then P \ H is p-centric in H and Fp(H)-radical. __ __ __ __ (d)If G_C G has_p-power_index, then P = G \ P for some P G which is p-centric in G and Fp(G )-radical. (e)If Q C G is a central p-subgroup, then Q P , and P=Q is p-centric in G=Q a* *nd Fp(G=Q)-radical. ff -1 (f)If eG-- i G is an epimorphism such that Ker(ff) Z(Ge), then ff (P ) is p-* *centric in eGand Fp(Ge)-radical. Proof.Point (a) follows from [JMO , Proposition 1.6(ii)]: P = P1xP2 for Pi Gi* *since P is p-radical, and Pi must be p-centric in Gi and Fp(Gi)-radical since CG(P ) = CG1(P1) x CG2(P2) and Out G(P ) ~=Out P1(G1) x OutP2(G2). Point (b) holds since CH (P ) CG(P ) and Op(Out H(P )) Op(Out G(P )). It remains to prove the other four points. (e) Fix a central p-subgroup Q Z(G). Then P Q, since otherwise 1 6= NQP (P )=P Op(NG(P )=P ). Also, P=Q is p-centric in G=Q, since otherwise the* *re would be x 2 Gr P of p-power order such that 1 6= [cx] 2 Ker OutG (P ) ---! OutG=Q(P=Q) x OutG(Q) Op(Out G(P )). It remains only to prove that P=Q is Fp(G=Q)-radical, and to do this it suffice* *s to show that Out G=Q(P=Q) ~=Out G(P ). Equivalently, since P=Q and P are p-centric, we must show that ___NG=Q(P=Q)_____~ NG(P ) = ___________0; C0G=Q(P=Q) x P=Q CG(P ) x P and this is clear once we have shown that C0G=Q(P=Q) ~=C0G(P ). _ 0 Any x 2 CG=Q(P=Q) lifts to an element x 2 G of order prime to p, whose conjugat* *ion action on P induces the identity on Q and on P=Q. By [Go , Corollary 5.3.3], al* *l such automorphisms of P have p-power order, and thus x centralizes P . Since Q is a * *p-group Ran Levi and Bob Oliver * * 49 and C0G=Q(P=Q) has order prime to p, this shows that the projection modulo Q se* *nds C0G=Q(P=Q) isomorphically to C0G(P ). ff -1 (f) Let eG- -i G be an epimorphism whose kernel is central. Clearly, ff P is* * p- centric in eG. It remains only to prove that ff-1P is Fp(Ge)-radical, and to do* * this it suffices to show that OutGe(ff-1P ) ~=Out G(P ). Equivalently, since P and ff-1(P ) are p-centric, we must show that ____NGe(ff-1P_)__~ NG(P ) = ___________0; C0eG(ff-1P ) x ff-1PCG(P ) x P and this is clear once we have shown that C0eG(ff-1P ) ~=C0G(P ). This follows by exactly the same argument as in the proof of (e). (c) Set P 0= P \ H for short. Let i 0 0 0 0 NH (P 0) --- --i Out H(P ) ~=NH (P )=(CH (P ).P ) be the natural projection, and set K = ß-1(Op(Out H(P 0))) NH (P 0). Then K Op(NH (P 0)) is an extension of CH (P 0).P 0by the p-group Op(Out H(P * *0)). It suffices to show that p - [K:P 0], since this implies that Op(Out H(P 0)) = 1 (* *i.e., P 0is Fp(H)-radical), and that any Sylow p-subgroup of CH (P 0) is contained in P 0(h* *ence P 0 is p-centric in H). fi Assume otherwise: that pfi[K:P 0]. Note first that P 0C NG(P ), and that NG(P* * ) NG(K); i.e., NG(P ) normalizes P 0and K. The first statement is obvious, and t* *he second is verified by observing directly that NG(P ) normalizes NH (P 0) and CH* * (P 0). Thus the action of NG(P ) on K induces an action of NG(P ), and in particularfo* *fiP , on K=P 0. Let K0=P 0denote the fixed subgroupfofithis action of P . Since pfi[K* *:P 0] by assumption, and since P is a p-group, pfi|K0=P 0|. A straightforward check also* * shows that K0 C NG(P ), and therefore that P K0 C NG(P ). Also, since P 0 K0 H, P K0=P ~=K0=(P \ K0) = K0=P 0 is a normal subgroup of NG(P )=P of order a multiple of p. Since P is p-centric* * in G by assumption, Out G(P ) = NG(P )=(CG(P ).P ) = NG(P )=(C0G(P ) x P ), and hence the image of P K0=P in OutG (P ) is a normal subgroup which also has * *order a multiple of p. By definition of K as an extension of CH (P 0).P 0by a p-group, if x 2 K has * *order prime to p, then x 2 CH (P 0). Hence if x 2 K0 has order prime to p, then for * *every z 2 P , [x, z] 2 P 0, so x acts trivially on P=P 0. Since x also centralizes P * *0, it follows that x centralizes P . This shows that the image of P K0=P in OutG (P ) is a p-group* *, thus a nontrivial normal p-subgroup of OutG (P ), and this contradicts the original as* *sumption that P is Fp(G)-radical. 50 Construction of 2-local finite groups __ (d) Let G C G be a normal subgroup of p-power index and let P G be a p-centr* *ic and Fp(G)-radical subgroup. Let i ffi N_G(P ) --- --i Out _G(P ) ~=N_G(P ) C_G(P ).P be the natural surjection, and set K = ß-1 Op(Out _G(P )) N_G(P ). __ Then K is an extension of C_G(P ).P by Op(Out _G(P )). Fix any P 2 Sylp(K). We * *will __ __ __ __ show that P \ G = P , and that P is p-centric in G and Fp(G )-radical. For each x 2 K \ G NG(P ), ß(x) 2 Op(Out _G(P )) \ OutG(P ) Op(Out G(P )) = 1. Hence 0 x 2 Ker NG(P ) ----! Out_G(P ) = (C_G(P ).P ) \ G = CG(P ) . P ~=CG(P ) x * *P, where C0G(P ) CG(P ) is of order prime to p. Since the_opposite inclusion_is * *obvious, this shows that K \ G = C0G(P ) x P , and hence (since P 2 Sylp(K)) that P \ G * *= P . __ Next, note that (K \_G) C K and K=(K_\ G) G =G, and hence K=C0G(P ) has * *__ p-power order. Since P 2 Sylp(K), P is an extension of P by K=(K \ G), and NK (* *P ) is an extension of a subgroup of_(K \ G) = (C0G(P ) x_P )_by_K=(K \ G). Also, * *an element x 2 C0G(P ) normalizes P if and only if [x, P] 2 P \ C0G(P ) = 1. Hence __ __ __ __ __ NK (P ) = CK (P ).P = C0G(P ) x P, (1) __ __ __ where_C0G(P ) = C0G(P ) \ CG(P ) has order_prime_to p_and is_normal in NK (P_).* *_Since C_G(P ) C_G(P ) K, (1) shows that C_G(P ) C0G(P ) x P, and hence that P * *is __ p-centric in G . __ __ It remains to show that P is Fp(G )-radical. Note first that K C N_G(P ) by c* *onstruc- __ tion, so for any x 2 N_G(P ), xP x-1 2 Sylp(K). Since K is an extension of C0G(* *P ) x P by the p-group_K=(K_\ G), and since C0G(P ) C K,_it follows_that K is a split e* *xtension of C0G(P ) by P . Hence for any x 2 N_G(P ), xP x-1 = yP y-1 for some y 2 C0G(* *P ). Consequently, the restriction map __ __ __ N_G(P )=C_G(P ) ~=AutG_(P ) ------! Aut _G(P ) ~=N_G(P )=C_G(P )(2) __ __ __ * * __ is surjective. Also, if x 2 C_G(P ) K normalizes P , then x 2 NK (P ) ~=P x C* *0G(P ) __ * * __ by (1), and so cx 2 Inn(P ). Thus the kernel of the map in (2) is contained in * *Inn(P ). Consequently, __ __ __ Out _G(P ) = Aut_G(P )= Inn(P ) ~=AutG_(P )= Aut_P(P ) ~=Out _G(P )=Op(Out _* *G(P )), __ __ and it follows that P is Fp(G )-radical. This is now applied to show the following: Proposition A.12. Fix an odd prime power q, and let P Spin7(q) be any sub- _ group which is 2-centric and F2(Spin7(q))-radical. Then P is centric in Spin7(* *F q); i.e., CSpin7(_Fq)(P ) = Z(P ). Ran Levi and Bob Oliver * * 51 Proof.Let z be the central involution in Spin7(q). By Lemma A.11(e), z 2 P , a* *nd __def P = P= is 2-centric in 7(q) and is F2( 7(q))-radical._So by Lemma A.11(d),* * there is a 2-subgroup bP O7(q) such that bP\ 7(q) = P , and such that bPis 2-centri* *c in O7(q) and is F2(O7(q))-radical. L m Let V = i=1Vi be a maximal decomposition of V as an orthogonal direct sum of bP-representations, and set bi = b|Vi. We assume these are arranged so that for* * some k, dim(Vi) > 1 when i k and dim(Vi) = 1 when i > k. Let V+ be the sum of those 1-dimensional components Viwith square discriminant, and let V- be the sum of t* *hose 1-dimensional components Viwith nonsquare discriminant. We will be referring to* * the two decompositions Mm Mk (V, b) = (Vi, bi) = (Vi, bi) (V+, b+) (V-, b-), i=1 i=1 both of which are orthogonal direct sums. We also write _ (1) _ V (1)= Fq FqV and Vi = Fq FqVi, and let b(1) and b(1)ibe the induced quadratic forms. Step 1: For each i, set Di= { IdVi} O(Vi, bi), a subgroup of order 2; and write Ym Y D = Di O(V, b), and D = Di O(V , b ). i=1 Vi V Thus D and D are elementary abelian 2-groups of rank m and dim(V ), respective* *ly. We first claim that bP D, (1) and that Ym Pb = Pi where 8 i, Pi is 2-centric in O(Vi, bi) and F2(O(Vi, bi))-radica* *l.(2) i=1 Clearly, [D, bP] = 1 (and D is a 2-group), so D bPsince bPis 2-centric. This * *proves (1). The Viare thus distinct (pairwise nonisomorphic) as bP-representations, si* *nce they are pairwise nonisomorphic as D-representations. The decomposition as a sum of * *Vi's is thus unique (not only up to isomorphism), since Hom bP(Vi, Vj) = 0 for i 6= * *j. Let bCbe the group of elements of O(V, b) which send each Vito itself, and le* *t bNbe the group of elements which permute the Vi. By the uniqueness of the decomposit* *ion of V , Ym bP.CO(V,b)(Pb) bC= O(Vi, bi) and NO(V,b)(Pb) bN. i=1 Since bPis 2-centric in O(V, b) and F2(O(V, b))-radical, it is also 2-centric i* *n Nb and F2(Nb)-radical (this holds for any subgroup which contains NO(V,b)(Pb)). So by * *Lemma A.11(b) (and since Cb C Nb), Pb is 2-centric in Cb and F2(Cb)-radical. Point (* *2) now follows from Lemma A.11(a). 52 Construction of 2-local finite groups Step 2: Whenever dim(Vi) > 1 (i.e., 1 i k), then by Lemma A.6, dim(Vi) is even, and bihas square discriminant. So by Lemma A.4(a), - IdVi2 (Vi, bi) for * *such i. Together with (1), this shows that __ Yk P = bP\ 7(q) Dix (V+, b+) \ D+ x (V-, b-) \ D- . (3) i=1 Also, by Lemma A.4(a) again, (V , b ) \ D = SO(V ,b ) \ D fi ff(4) = - IdVi Vjfik+ 1 i < j m, Vi, Vj V . __ Step 3: By (3) and (4), the Vi are distinct as P -representations (not only a* *s Pb- representations), except possibly when dim (V ) = 2. We first check that this * *ex- ceptional case_cannot_occur. If dim (V+) = 2 and its_two irreducible summands * *are isomorphic as P -representations, then the image of P under projection to O(V+,* * b+) is_just { IdV+}. Hence we can write V+ = W W 0, where W ?W 0are 1-dimensiona* *l, P-invariant, and have nonsquare discriminant. Also, dim(V-) is odd,_since V+ an* *d_the_ Vifor i k are all even dimensional._So - IdV- W lies in C 7(q)(P ) but not in* * P. But this is impossible, since P is 2-centric in 7(q). The argument when dim(V-) = * *2 is similar. __ The Vi are thus distinct as P -representations. So for all i 6= j, Hom P (Vi* *, Vj) = 0, and hence _ Hom _Fq[P](Vi(1), Vj(1)) ~=Fq FqHom Fq[P](Vi, Vj) = 0. __ (1) Thus any element of O(V (1), b(1)) which centralizes P sends each Vi to itsel* *f. In other words, __ Ym (1) (1) CSpin7(_Fq)(P )= C 7(_Fq)(P ) O(Vi , bi ). i=1 __ If dim (V ) 2, then since P contains all involutions in O(V , b ) which ar* *e Pb- invariant and have even dimensional_(-1)-eigenspace (see (3)), Lemma A.4(c) sho* *ws that each element of Spin7(F q)_which commutes with P must act on V via Id. * *Also, for 1 i k, since - IdVi2 P by (3), each element in the centralizer of P act* *s on Vi with determinant 1 (Lemma A.4(c) again). We thus conclude that Yk CSpin7(_Fq)(P )= SO(Vi(1), b(1)i) x { IdV+} x { IdV-}.(5) i=1 Step 4: We next show that Yk CSpin7(_Fq)(P )= { IdVi} x { IdV+} x { IdV-}. (6) i=1 Using (5), this means showing, for each 1 i k, that pri CSpin7(_Fq)(P )= { IdVi}; (7) _ (1) (1) where pri denotes the projection of O7(F q) = O(V (1), b(1)) to O(Vi , bi ). * * By Lemma A.6, dim(Vi) = 2 or 4. We consider these two cases separately. Ran Levi and Bob Oliver * * 53 Case 4A: If dim(Vi) = 4, then by (2) and Lemma A.11(c), Pi0def=Pi\ (Vi, bi) is 2-centric in (Vi, bi) and is F2( (Vi, bi))-radical. Also, by Proposition A.5, (Vi, bi) ~= +4(q) ~=SL2(q) xC2 SL2(q). By Lemma A.11(a,f), under this identification, we have Pi0= QxC2Q0, where Q and* * Q0 are 2-centric in SL2(q) and F2(SL2(q))-radical. The Sylow 2-subgroups of SL2(q)* * are quaternion groups of order 8, all subgroups of a quaternion 2-group are quate* *rnion or cyclic, and cyclic 2-subgroups of SL2(q) cannot be both 2-centric and F2(SL2* *(q))- radical. So Q and_Q0 must be quaternion of order 8. By [Sz, 3.6.3], any cyc* *lic 2-subgroup of SL2(F q) of order 4 is conjugate to a subgroup of_diagonal matr* *ices, whose centralizer is the group of all diagonal matrices in SL2(F q)._ Knowing t* *his, one easily_checks that all nonabelian quaternion 2-subgroups of SL2(F q) are ce* *ntric in SL2(F q). It follows that Pi0is centric in _ _ SO(Vi(1), b(1)i) ~=SL2(F q) xC2 SL2(F q), and hence that 0 0 priCSpin7(_Fq)(P )= CSO(V (1) (1)(Pi) = Z(Pi) = { IdVi}. i ,bi ) Thus (7) holds in this case. Case 4B: If dim(Vi) = 2, then O(Vi, bi) ~=O2 (q) is a dihedral group of order * *2(q 1) [Ta , Theorem 11.4]. Hence Pi 2 Syl2(O(Vi, bi)), since the Sylow subgroups are* * the only radical 2-subgroups of a dihedral group. Fix Vj for any k < j m, and cho* *ose ff 2 O(Vi, bi) of determinant (-1) whose (-1)-eigenspace has the same discrimin* *ant as Vj. Since Pi 2 Syl2(O(Vi, bi)), we_can assume (after conjugating if necessar* *y) that ff 2 Pi. Then (- IdVj) ff lies in P = bP\ 7(q). Hence for any g 2 CSpin7(_Fq* *)(P )=, pri(g) 2 O(Vi(1), b(1)i) leaves both eigenspaces of ff invariant, and has deter* *minant 1 by (5). Thus pri(g) = IdVi; and so (7) holds in this case. Step 5: Clearly, - IdV lies in SO(V , b ) if and only if dim(V ) is even (whi* *ch is the case for exactly one of the two spaces V ), and this holds if and only if -* * IdV 2 (V , b ). Also, since each Vi for 1 i k has square discriminant (Lemma A.6 again), - IdVi2 (Vi, bi) for all such i. Thus (6) and (1) imply that __ CSpin7(_Fq)(P )= bP\ 7(q) = P, _ and hence that P is centric in Spin7(F q). Proposition A.12 does not hold in general if Spin7(-) is replaced by an arbit* *rary algebraic group. For example, assume q is an odd prime power, and let P SL5(q) be the group of diagonal matrices of 2-power order. Then P is_2-centric in SL5(* *q) and F2(SL5(q))-radical, but is definitely not 2-centric in SL5(F q). References [As1] M. Aschbacher, A characterization of Chevalley groups over fields of odd * *order, Annals of Math. 106 (1977), 353-398 [As2] M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986) [Be] D. 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Notbohm, On the 2-compact group DI(4) (preprint) [Pu] L. Puig, Unpublished notes [Sm] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, * *Trans. Amer. Math. Soc. 129 (1967), 58-93 [Sm2] L. Smith, Polynomial invariants of finite groups, A. K. Peters (1995) [So] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 2* *8 (1974), 182-198 [Sz] M. Suzuki, Group theory I, Springer-Verlag (1982) [Ta] D. Taylor, The geometry of the classical groups, Heldermann Verlag (1992) [Wb] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press * *(1994) [Wi] C. Wilkerson, A primer on the Dickson invariants, Proc. Northwestern homo* *topy theory conference 1982, Contemp. Math. 19 (1983), 421-434 Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, U.K. E-mail address: ran@maths.abdn.ac.uk LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France E-mail address: bob@math.univ-paris13.fr