STRONGLY CLOSED SUBGROUPS AND THE CELLULAR STRUCTURE OF CLASSIFYING SPACES RAM'ON J. FLORES AND RICHARD M. FOOTE Abstract. In this paper we give a complete classification of the finite g* *roups that contain a strongly closed p-subgroup, generalizing previous work of * *the second author to the case of an odd prime. We use this result to also obtain a d* *escription of the BZ=p-cellularization (in the sense of Dror-Farjoun) of all the classi* *fying spaces of finite groups. 1.Introduction This paper brings to fruition previous endeavors of the two authors working i* *n- dependently in two areas of Mathematics _ Finite Group Theory and Topology _ by completing both the classification of finite groups containing a strongly cl* *osed p-subgroup for every prime p, and the characterization of cellularization of cl* *assify- ing spaces of all finite groups. These two classifications, the latter relying* * on the former, epitomize the rich interplay between their subject areas that has histo* *ri- cally been evident and is currently even more vibrant. The results also mirror* * the striking advances in topics such as fusion systems, that have recently captured* * the interest of both topologists and group theorists. It is interesting that in the* * precur- sor to this paper, [FS07 ], the germs of the ideas for characterizing classifyi* *ng spaces were already present. However, the list of simple groups containing strongly cl* *osed p-subgroups for p odd, characterized herein, is much more diverse; and indeed it uncovered new "obstructions" that had to be dealt with. In particular, one infi* *nite family exhibits a unique fusion behavior that persists even in under more strin* *gent (generation) conditions than just strong closure. Thus we see the necessity of * *having the full group-theoretic classification in order to effect the complete topolog* *ical solu- tion. Finally, although the techniques used in the two classifications tend to * *be quite different, the underlying fusion arguments that permeate the group theory secti* *ons ___________ Date: July 5, 2007. The first author was supported by MEC grant MTM2004-06686. 1 2 RAM'ON J. FLORES AND RICHARD M. FOOTE provide deeper insight into, in essence, the fusion that may be "swept under th* *e rug" for our topological considerations (in a sense to be made precise shortly). Ind* *eed, the marriage of these elements is seen in high relief in Section 5 where we explore* * more explicit configurations that give rise to interesting _ what might be called ex* *otic _ classifying spaces. We now give the necessary terminology and some very brief historical contextu- alization before stating the main results. Recall that for any finite group G * *and subgroup S we say x, y 2 S are fused if they are conjugate in G (but not necess* *arily in S). This concept has played a central role in group theory and representati* *on theory, particularly in the case when S is a Sylow p-subgroup of G. Of particu* *lar relevance to our work are the celebrated Glauberman Z*-Theorem, [Gla66 ], and t* *he Goldschmidt Theorem on strongly closed abelian 2-subgroups, [Gol74 ]. A subgroup A of S is called strongly closed in S with respect to G if for every a 2 A, eve* *ry element of S that is fused in G to a lies in A; in other words, aG \ S A, where aG de* *notes the G-conjugacy class of a. It is easy to verify that if A is a p-subgroup, the* *n A is strongly closed in a Sylow p-subgroup if and only if it is strongly closed in N* *G (A), so the notion of strong closure for a p-subgroup does not depend on the Sylow subg* *roup containing it. For a p-group A we therefore simply say A is strongly closed. * *The __ __ Z*-Theorem proved that if A is strongly closed and of order 2, then A Z(G ), * *where the overbars denote passage to G=O20(G). Goldschmidt extended this by showing ___G that if A is a strongly closed abelian 2-subgroup, then < A > is a central pro* *duct of an abelian 2-group and quasisimple groups that have either BN-rank 1 or abelian Sylow 2-subgroups. These two theorems, in particular, played fundamental roles * *in the study of finite groups, especially in the Classification of the Finite Simp* *le Groups. The concept of strong closure has had ramifications even beyond finite group * *theory, as our work will illustrate (see also [Foo97a ], for example). Furthermore, obs* *erve that if A S is strongly closed, NG (S) also normalizes A; in other words, every fu* *sion automorphism of S restricts to an automorphism of A. This important property has been used to extend the notion of strong closure to more general frameworks. For example, in the context of p-local finite groups and linking systems ([BLO0* *3 ]) the adequate concept of strong closure has been crucial in the study of extensi* *ons ([BCGLO07 ]), and in the definition and structure theory of saturated fusion * *systems ([Lin06] and [Asc07 ]). CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 3 In this paper we are essentially concerned with the relationship between the * *(group- theoretic) notion of strong closure and the concept of cellular classes in Homo* *topy Theory. Recall, following Dror-Farjoun [Far95], that given pointed spaces A and X, X is said to be A-cellular if it can be built from A by iterating pointed ho- motopy colimits. In a natural way, one may define an A-cellularization functor CW A : Spaces *! Spaces *, that is idempotent and augmented, with the property that for every X the space CW AX is A-cellular, and the augmentation CW AX ! X induces a weak homotopy equivalence: map *(A, CW AX) ' map *(A, X). Moreover, the cellularity class of A is defined as the minimal class of spaces that conta* *ins all the A-cellular spaces. In this way the category of pointed spaces is divided in* *to cel- lularity classes. All these concepts were formalized by Chach'olski [Cha96 ], a* *nd they have been recently applied in very different contexts such as the theory of alg* *ebraic varieties, commutative rings, stable homotopy and, more generally, Duality Theo* *ry ([DGI01 ]). In this paper we classify, for every prime p, the finite groups possessing a * *strongly closed p-subgroup. This enables us to give a complete description of the BZ=p- cellularization of classifying spaces of all finite groups. The philosophy beh* *ind our work is the following: whenever one has a space X with a notion of p-fusion (and then strong closure), the knowledge of the strongly closed subobjects of X is d* *eeply related (and in some cases, almost equivalent) to the A-cellular structure of X* *, for a certain p-torsion space A. This strategy opens up new perspectives to analyze (* *from the point of view of (Co)localization Theory) the p-primary structure of a wide* * class of homotopy meaningful spaces, such as p-local finite groups, classifying space* *s of compact Lie groups, p-compact groups or, more generally p-local compact groups ([BLO07 ]). To describe the main results we introduce some new notation. Henceforth p is * *any prime, S is a Sylow p-subgroup of the finite group G and A is a subgroup of S. * *It is completely elementary that there is a unique normal subgroup of G, denoted by OA(G), that is maximal with respect to the property that A contains one of its * *Sylow p-subgroups, i.e., A \ OA(G) 2 Sylp(OA(G)). Note that A OA(G) if and only if A is a Sylow p-subgroup of its normal closure < AG > in G. In the latter circumst* *ance A is strongly closed, i.e., Sylow p-subgroups of normal subgroups of G are the "g* *eneric" instances of strongly closed p-subgroups. One may therefore view our classifica* *tion as a determination of the "obstructions" to this generic reason that strongly c* *losed 4 RAM'ON J. FLORES AND RICHARD M. FOOTE p-subgroups arise. In what follows let overbars denote passage to G=OA(G), so t* *hat __ __ A does not contain a Sylow p-subgroup of any nontrivial normal subgroup of G . * *The complete description of groups possessing a strongly closed p-subgroup, Theorem* *s 2.1 and 2.2, is too lengthy and technical to warrant interrupting the flow of this * *intro- duction, so _ in the spirit of our cursory statement of Goldschmidt's Theorem _* * we present only an abridged version at this point: Theorem A. Let p be any prime, let G be a finite group possessing a strongly cl* *osed __ p-subgroup A and assume A is not a Sylow p-subgroup of < AG >. Then A 6= 1 and ___G < A > = (L1 x L2 x . .x.Lr) where r 1, each Li is a simple group belonging to an explicitly listed family* *, Ai = __ A \ Li is a homocyclic abelian group, and is a (possibly trivial) group actin* *g as automorphisms on each Li. This statement combines the p = 2 case (Theorem 2.1) and the p odd case (The- orem 2.2). In addition to the families of simple groups being explicitly liste* *d, all __ possibilities for each Ai are given, as is the precise action of on each Li. * *A crucial consequence of this theorem is the following: Corollary B. Assume the hypothesis of Theorem A, let A S 2 Sylp(G), and __ assume that G is generated by conjugates of A. Then N__G(A ) controls strong fu* *sion __ in S. This classification was the main ingredient we lacked in order to finish the * *charac- terization of CW BZ=pBG for all finite groups G, solving a problem that was po* *sed by Dror-Farjoun [Far95, 3.C] in the case G = Z=pr, and partially solved in [Flo* *07] and [FS07 ] (see Section 4.1 below for an analysis of the previous cases). The * *latter paper showed the importance of a specific strongly closed subgroup: for S a Syl* *ow p-subgroup of G let A1(S) denote the unique minimal strongly closed subgroup of* * S that contains all elements of order p in S (i.e., contains 1(S), the group gen* *erated by elements of order p in S _ sometimes called the p-socle of S). The first ma* *in result on classifying spaces is the following. Theorem C. Let G be a finite group generated by its elements of order p, let A = A1(S), and assume A 6= S. Let overbars denote passage from G to the quotient gr* *oup CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 5 G=OA(G). Then there exists a fibration __ __^ CW BZ=p(BG^p) -! BG^p-! B(N__G(A )= A)p. This is Theorem 4.2. It makes precise what we glibly called "sweeping under t* *he rug" earlier. It is important to note that although it is relatively elementar* *y to prove that for any strongly closed subgroup A the subgroup NG (A) controls fusi* *on of subgroups containing A, it does not generally control fusion inside A (or for s* *ubsets that intersect A but do not contain it). Thus the precise "shape" of the classi* *fication _ factoring out the "correct" subgroup OA(G) combined with an explicit knowledge of the structure of the quotient G=OA(G) _ is crucial. Furthermore, when G=OA(G) is simple and equal to one of the "obstruction groups" Liof Theorem A, in all b* *ut one __ family the "smaller" group N__G(S ) controls strong fusion in S (and this was a* *lways the situation for p = 2); however, in that one exceptional family this replacem* *ent is not possible, as we explicate in Section 5. To describe the culminating main result let 1(G) denote the subgroup of G ge* *ner- ated by the elements of order p in G (where p is always clear from the context)* *, and let BG^pdenote Bousfield-Kan p-completion of BG. For the definition and propert* *ies of this functor, we refer the reader to [BK72 ] for a thorough account and to [* *Flo07, Section 2 ] for a brief survey. By 4.1 and 4.3 of the latter, the inclusion 1(* *G) ,! G induces a homotopy equivalence CW BZ=pB 1(G) ' CW BZ=pBG. Thus the follow- ing result (which is Theorem 4.3 below and combines the information obtained in [Flo07], [FS07 ] and the present article), gives all the possible homotopy stru* *ctures for CW BZ=pBG: Theorem D. Let G be a finite group generated by its elements of order p, let S 2 Sylp(G), and let A = A1(S) be the minimal strongly closed subgroup of S contain* *ing 1(S). Then the BZ=p-cellularization of BG has one the following shapes: (1) If G = S is a p-group then BG is BZ=p-cellular. (2) If G is not a p-group and A = S then CW BZ=pBG is the homotopy fiber of Q the natural map BG ! q6=pBG^q. (3) If G is not a p-group and A 6= S then CW BZ=pBG is the homotopy fiber of __ __ Q the map BG ! B(N__G(A )= A)^px q6=pBG^q. 6 RAM'ON J. FLORES AND RICHARD M. FOOTE The classification of groups containing a strongly closed p-subgroup gives a * *very precise description of the fiber of the augmentation CW BZ=pBG ! BG in terms of normalizers of strongly closed p-subgroups in the simple components of G=OA(G). In this sense the results here further improve those of [FS07 ], where this deg* *ree of sharpness was only obtained in the description of some concrete examples. The overall organization of the paper is as follows: Section 2 begins by rec* *apit- ulating the basic terminology and previous results in group theory; it then sta* *tes the main group-theoretic classification in detail. After some preliminary resul* *ts the Sylow structure and Sylow normalizers of simple groups containing strongly clos* *ed p-subgroups are elucidated. The main group-theoretic theorem is derived at the * *end of this section as a consequence of an inductive special case describing the "m* *ini- mal configuration" on groups containing a minimal strongly closed p-subgroup, p* * odd (Theorem 2.3). Section 3 consists of the proof for this minimal configuration.* * In Section 4 we provide additional background and more precise statements of previ* *ous results of a topological nature, and then the main results on cellularization a* *re es- tablished. Section 5 illustrates the efficacy of our methods by describing NG (* *A) and NG (S) as well as computing CW BZ=pBG for specific cases of G. More explicitly* *, we describe first CW BZ=pBG for G simple, and then for split extensions, and fina* *lly for certain nonsplit extensions of simple groups. The latter are very illuminating * *in the sense that they give an alluring glimpse of what "should be" the BZ=p-cellulari* *zation of more general objects. Acknowledgements. We thank Carles Broto, David Dummit, Bob Oliver and J'er^ome Scherer for helpful discussions, and also George Glauberman for provid* *ing a motivating example. We thank Michael Aschbacher for sharing his lecture notes, * *and acknowledge that a number of the results in Section 3 were also proven independ* *ently by him. 2. Strongly closed p-subgroups Throughout this section G is a finite group, p is a prime and A is a p-subgro* *up of G. Following the definitions and description in the Introduction, this section * *and the next complete the classification of groups possessing a strongly closed p-subgr* *oup by carrying out for all odd p the classification scheme that was done in [Foo97 ] * *for p = 2 (see Theorem 2.1 below). CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 7 In general let R be any p-subgroup of G. If N1 and N2 are normal subgroups of* * G with R\Ni2 Sylp(Ni) for both i = 1, 2, then R\N1N2 is a Sylow p-subgroup of N1N* *2. Thus there is a unique largest normal subgroup N of G for which R \ N 2 Sylp(N); denote this subgroup by OR(G). Thus R is a Sylow p-subgroup of < RG > if and onlyRif OR(G). Note that Op0(G=OR(G)) = 1; in particular, if R = 1 is the identity subgroup th* *en O1(G) = Op0(G). In general, ROR(G)=OR(G) does not contain the Sylow p-subgroup __ of any nontrivial normal subgroup of G=OR(G); in other words, O__R(G ) = 1, whe* *re overbars denote passage to G=OR(G). Throughout this section we freely use the observation that strong closure passes to quotient groups (cf. Lemma 2.8), so w* *hen analyzing groups where R 6 OR(G) we may factor out OR(G). With this in mind, the classification for strongly closed 2-subgroups from [Foo97 ] is as follows: Theorem 2.1. Let G be a finite group that possesses a strongly closed 2-subgrou* *p A. __ __ Assume A is not a Sylow 2-subgroup of < AG >, and let G = G=OA(G). Then A 6= 1 ___G and < A > = L1 x L2 x . .x.Lr, where each Li is isomorphic to U3(2ni) or Sz(2n* *i) __ for some ni, and A \ Li is the center of a Sylow 2-subgroup of Li. The classification for p odd, which is the principal objective of this sectio* *n, yields a more diverse set of "obstructions" with added "decorations" as well. Theorem 2.2. Let p be an odd prime and let G be a finite group that possesses a strongly closed p-subgroup A. Assume A is not a Sylow p-subgroup of < AG >, and* * let __ __ G = G=OA(G). Then A 6= 1 and ___G (2.1) < A > = (L1 x L2 x . .x.Lr)(D . AF) __ where r 1, each Liis a simple group, and Ai= A \Liis a homocyclic abelian gro* *up. Furthermore, D = [D, AF] is a (possibly trivial) p0-group normalizing each Li, * *and AF __ is a (possibly trivial) abelian subgroup of A of rank at most r normalizing D a* *nd each Li and inducing outer automorphisms on each Li, and the extension (A1. .A.r) : * *AF splits. Each Li belongs to one of the following families: (i)Li is a group of Lie type in characteristic 6= p whose Sylow p-subgroup * *is abelian but not elementary abelian; in this case the Sylow p-subgroup of* * Li is homocyclic of the same rank as Ai but larger exponent than Ai; here D=(D* * \ 8 RAM'ON J. FLORES AND RICHARD M. FOOTE LiC__G(Li)) is a cyclic p0-subgroup of the outer diagonal automorphism g* *roup of Li, and AF=CAF (Li) acts as a cyclic group of field automorphisms on * *Li. (ii)Li~= U3(pn) or Re(3n) is a group of BN-rank 1 (p = 3 with n odd and 2 * *in the latter family); in the unitary case Aiis the center of a Sylow p-sub* *group of Li(elementary abelian of order pn), and in the Ree group case Aiis eithe* *r the center or the commutator subgroup of a Sylow 3-subgroup (elementary abel* *ian of order 3n or 32n respectively); in both families D and AF act triviall* *y on Li. (iii)Li~= G2(q) with (q, 3) = 1; here |Ai| = 3 and both D and AF act triviall* *y on Li. (iv) Li is one of the following sporadic groups, where in each case Ai has pr* *ime order, and both D and AF act trivially on Li: (p = 3) :J2, (p = 5) :Co3, Co2, HS, Mc, (p = 11):J4. (v) Li ~=J3, p = 3, and Ai is either the center or the commutator subgroup o* *f a Sylow 3-subgroup (elementary abelian of order 9 or 27 respectively); her* *e D and AF act trivially on Li. Remark. After factoring out OA(G) _ so that overbars may be omitted _ the proof of Theorem 2.2 shows that F *(G) = L1 x . .x.Lr, and (2.1) may also be written * *as < AG > ~=((L1 x . .x.Li)D x Li+1x . .x.Lj)AF x (Lj+1x . .x.Lr) where L1, . .,.Liare the components of type P SL or P SU over fields of charact* *eristic 6= p, Li+1, . .,.Lj are other groups listed in conclusion (i) (but not linear o* *r unitary), and Lj+1, . .,.Lr are the components of types listed in (ii) to (v). Furthermo* *re, assume G = < AG > and let A S 2 Sylp(G) and S* = S \ F *(G). Then we may choose D generically as [Op0(CG (S*)), S], which is a p0-group normalized by S * *and centralized by the Sylow p-subgroup S* of L1. .L.r. Conversely, observe that any finite group that has a composition factor of on* *e of the above types for Li possesses a strongly closed p-subgroup that is not a Syl* *ow p-subgroup of its normal closure in G. More detailed information about the stru* *cture of the Sylow p-subgroups and their normalizers for the simple groups Liappearin* *g in the conclusion to this theorem is given from Proposition 2.9 through Corollary * *2.13 following. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 9 Theorem 2.2 is derived at the end of this section as a consequence of the nex* *t result, which is the minimal configuration whose proof appears in the next section. Theorem 2.3. Assume the hypotheses of Theorem 2.2. Assume also that A is a minimal strongly closed subgroup of G, i.e., no proper, nontrivial subgroup of * *A is also strongly closed. Then the conclusion of Theorem 2.2 holds with the additi* *onal results that A is elementary abelian, D = 1, AF = 1, and G permutes L1, . .,.Lr transitively (hence they are all isomorphic). Although these group-theoretic results are of independent interest, the impor* *tant consequences we need for our main theorems on cellularization are the following* * _ their proofs appears at the end of this section. Corollary 2.4. Let p be any prime, let G be a finite group containing a strongl* *y closed __ p-subgroup A, let S be a Sylow p-subgroup of G containing A, and let G = G=OA(G* *). __ * * __ Assume that G is generated by the conjugates of A. Then N__G(A ) controls stron* *g G - __ __ fusion in S. Furthermore, if p 6= 3 or if G does not have a component of type G* *2(q) fi __ __ __ with 9 fiq2 - 1, then N__G(S ) controls strong G -fusion in S. In Section 5.3 we demonstrate that the exceptional case to the stronger concl* *usion in the last sentence of Corollary 2.4 is unavoidable, even if we impose the con* *dition that 1(S) A: we construct examples of groups G generated by conjugates of a __ strongly closed subgroup A containing 1(S) and G=OA(G) ~= G2(q) where N__G(S ) __ does not control fusion in S. The next result facilitates computation of NG (A) in groups satisfying the co* *nclusion to the preceding corollary. Corollary 2.5. Assume the hypotheses of preceding corollary and the notation of __ Theorem 2.2. For each i let Ci= C__G(AF) \ NLi(Ai) and Si= S \ Li. Then __ __ N__G(A )= A = (S1C1=A1) x (S2C2=A2) x . .x.(SrCr=Ar). In particular, if Li is a component on which AF acts trivially _ which is the c* *ase for all components in conclusions (ii) to (v) of Theorem 2.2 _ the ithdirect factor* * above may be replaced by just NLi(Ai)=Ai (and this applies to all factors if AF = 1). Example. An example where both D and AF are nontrivial is G = P GL11(q)< f > with p = 5 and q = 35: Here the simple group L = P SL11(q) has an abelian Sylow 10 RAM'ON J. FLORES AND RICHARD M. FOOTE 5-subgroup of type (25,25), P GL11(q)=L is the cyclic outer diagonal automorphi* *sm group of L of order 11 (this is DL=L), and < f > = AF induces a group of order 5 of field automorphisms on P GL11(q); in particular, G=L is the non-abelian group of order 55. If f 2 S 2 Syl5(G), then A = 1(S) = < f, 1(S \ L) > is elementa* *ry abelian of order 53 and strongly closed in S with respect to G, and A* = 1(S \* * L) is a minimal strongly closed subgroup of G. In this example, to compute the normalizers of A and A* it is easier to work * *in the universal group GL11(q)< f > _ also denoted by G _ via its action on an 11- dimensional Fq-vector space V (since the center of GL11(q) has order prime to 5* *) _ see the proof of Lemma 3.4 for some general methodology. Let G* = GL11(q) and S* = S \ G*. Then one sees that NG (A*) = NG (S*) is contained in a subgroup H = ((G1 x G2)< t > x C)< f > where Gi ~=GL4(q), C ~=GL3(q), t interchanges the two factors and f induces fie* *ld automorphisms on all three factors and commutes with t (here G1 x G2 x C acts naturally on a direct sum decomposition of V ). Let Si = S \ Gi, so Si is cycl* *ic of order 25 and acts Fq-irreducibly on the 4-dimensional submodule for Gi. By bas* *ic representation theory, CGi(Si) is cyclic of order q4 - 1, and NGi(Si)=CGi(Si) i* *s cyclic of order 4. Thus NG (A*) = NG (S*) ~=((q4 - 1) . 4 x (q4 - 1) . 4)< t, f > x GL3(35). Since AF = < f > acts as a field automorphisms, similar considerations show that NG (A) = S(NG (S*) \ CG*(f)) ~=(400 . 4 x 400 . 4)< t, f > x GL3(3). The G-fusion in S is effected by the group S(4 x 4)< t >, which is the same for* * both normalizers. In this example we may choose D = [CG (S*), f], which is of type B x B x (SL3(q) . 121) where B is cyclic of order (q4- 1)=5(34- 1); a (smaller)* * group of diagonal automorphisms for D could be chosen inside the abelian factor B x B. The proof of Theorem 2.3 relies on the Classification of Finite Simple Groups* *. We reduce to the case where a minimal counterexample, G, is a simple group having a strongly closed p-subgroup A that is properly contained in a non-abelian Sylow p-subgroup S of G. The remainder of the proof involves careful investigation of* * the families of simple groups to determine precisely when this happens. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 11 We note that "most" simple groups do possess a strongly closed p-subgroup tha* *t is proper in a Sylow p-subgroup, that is, conclusion (i) of Theorem 2.2 is the "ge* *neric obstruction" in the following sense. Let Ln(q) denote a simple group of Lie typ* *e and BN-rank n over the finite field Fq with (q, p) = 1. As we shall see in Section * *2.1, for all but the finitely many primes dividing the order of the Weyl group of the un* *twisted version of Ln(q) the Sylow p-subgroups of Ln(q) are homocyclic abelian. Further* *more, the order of Ln(q) can be expressed as a power of q times factors of the form * *m (q)rm for various m, rm 2 N, where m (x) is the mth cyclotomic polynomial. Then by Proposition 2.9 below, if m0 is the multiplicative order of q (mod p), then p d* *ivides m0(q) and the abelian Sylow p-subgroup of Ln(q) is homocyclic of rank rm0 and fi exponent | m0(q)|p. In particular it is not elementary abelian whenever p2 fi m* *0(q). For example, this is the case in the groups P SLn+1(q) whenever p > n + 1 and p2 divides qm - 1 for some m n + 1. Thus for fixed n and all but finitely many p* *, this can always be arranged by taking q suitably large. 2.1. Preliminary Results. The special case when A has order p has already been treated in [GLSv3 , Proposition 7.8.2]. It is convenient to quote this special* * case, although with extra effort our arguments could be reworded to independently sub* *sume it. Proposition 2.6. If K is simple and G = AK is a subgroup of Aut(K) such that A is strongly closed and |A| = p, then A K = G and either the Sylow p-subgroups of G are cyclic, or G is isomorphic to U3(p) or one of the simple groups listed* * in conclusions (iii) and (iv) of Theorem 2.2. The authors of this result remark that an immediate consequence of this is the odd-prime version of Glauberman's celebrated Z*-Theorem. Proposition 2.7. If an element of odd prime order p in any finite group X does * *not commute with any of its distinct conjugates then it lies in Z(X=Op0(X)). We record some basic facts about strongly closed subgroups (the second of whi* *ch relies on the odd-prime Z*-Theorem). Lemma 2.8. For p any prime let A be a strongly closed p-subgroup of G. (1) If N is any normal subgroup of G then AN=N is a strongly closed p-subgro* *up of G=N. 12 RAM'ON J. FLORES AND RICHARD M. FOOTE (2) If A normalizes a subgroup H of G with Op0(H) = 1 and A \ H = 1 then A centralizes H. Proof.In part (1) let A S 2 Sylp(G). This result follows immediately from the definition of strongly closed applied in the Sylow p-subgroup SN=N of G=N toget* *her with Sylow's Theorem. The proof of (2) is the same as for p = 2 since, as noted earlier, the Z*-Theorem holds also for odd primes: by induction reduce to the c* *ase where G = AH and CA(H) = 1. Then any element of order p in A is isolated, hence lies in the center. The next few results gather facts about the simple groups appearing in the co* *nclu- sions to Theorems 2.1 and 2.2. The cross-characteristic Sylow structures of the simple groups of Lie type gr* *oups are beautifully described in [GL83 , Section 10] and reprised in [GLSv3 , Secti* *on 4.10]. Let L(q) denote a universal Chevalley group or twisted variation over the field* * Fq. (In the notation of [GLSv3 ], L(q) = dL(q), where d = 1, 2, 3 corresponds to the un* *twisted, Steinberg twisted, or Suzuki-Ree twisted variations respectively). Let W denote* * the Weyl group of the untwisted group corresponding to L(q). Except for some small order exceptions, L(q) is a quasisimple group; for example A`(q) ~= SL`+1(q) and 2A`(q) ~=SU`+1(q). There is a set O(L(q)) of positive integers, and "multiplici* *ties" rm for each m 2 O(L(q)), such that Y |L(q)| = qN ( m (q))rm m2O(L(q)) where m (x) is the cyclotomic polynomial for the mthroots of unity. Let p be an odd prime not dividing q and assume S is a nontrivial Sylow p-sub* *group fi of L(q). Let m0 be the smallest element of O(L(q)) such that p fi m0(q). Let X (2.2) W = {m 2 O(L(q)) | m = pam0, a 1} and b = rm m2W where b = 0 if W = ;. The main structure theorem is as follows. Proposition 2.9. Under the above notation the following hold: (1) m0 is the multiplicative order of q (mod p). (2) Except in the case where L(q) = 3D4(q) with p = 3, S has a nontrivial no* *rmal homocyclic subgroup, ST, of rank rm0 and exponent | m0(q)|p. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 13 (3) With the same exception as in (2), S is a split extension of ST by a (po* *ssibly trivial) subgroup SW of order pb (where b is defined in (2.2)), and SW * * is fi * * fi isomorphic to a subgroup of W . In particular, if p 6 fi|W | or if pm0 6* * fim for all m 2 O(L(q)), then S = ST is homocyclic abelian. (4) If L(q) = 3D4(q) with p = 3 and |q2 - 1|3 = 3a, then S is a split extens* *ion of an abelian group of type (3a+1, 3a) by a group of order 3, and S has ran* *k 2. (5) If L(q) is a classical group (linear, unitary, symplectic or orthogonal)* * then every element of order p is conjugate to some element of ST. (6) Except in 3D4(q) (where SW is not defined), SW acts faithfully on ST; * *and in _____ ___ the simple group L(q)=Z(L(q)) = L(q) we have SW ~=SW acts faithfully on ___ fi fi ST except when p = 3 with L(q) ~=SL3(q) (with 3 fiq - 1 but 9 6 fiq - 1)* * or fi fi SU3(q) (with 3 fiq + 1 but 9 6 fiq + 1). (7) If a Sylow p-subgroup of the simple group L(q)=Z(L(q)) is abelian but not elementary abelian then p does not divide the order of the Schur multipl* *ier of L(q). Proof.For parts (1) to (6) see [GL83 , 10-1, 10-2] or [GLSv3 , Theorems 4.10.2,* * 4.10.3]. If the odd prime p divides the order of the Schur multiplier of L(q) then by [G* *LSv3 , Table 6.12] we must have L(q) of type SLn(q), SUn(q), E6(q) or 2E6(q) with p di* *viding (n, q - 1), (n, q + 1), (3, q - 1) or (3, q + 1) respectively. It follows easil* *y from (6) that in each of the corresponding simple groups a Sylow p-subgroup cannot be abelian* * of exponent p2. We shall frequently adopt the efficient shorthand from the sources just cited* * for the latter families. Notation. Denote SLn(q) by SL+n(q) and SUn(q) by SL-n(q) (likewise for the gene* *ral linear and projective groups); and say a group is of type SLffln(q) according t* *o whether fi p fiq - ffl for ffl = +1, -1 respectively (dropping the "1" from 1). The anal* *ogous convention is adopted for E6(q) = E+6(q) and 2E6(q) = E-6(q). The following general result is especially important for the groups of Lie ty* *pe. Proposition 2.10. If G is any simple group with an abelian Sylow p-subgroup S for any prime p, then NG (S) acts irreducibly and nontrivially on 1(S), and so* * S is homocyclic. In particular, a nontrivial subgroup of S is strongly closed if and* * only if it is homocyclic of the same rank as S. 14 RAM'ON J. FLORES AND RICHARD M. FOOTE Proof.See [GLSv3 , Proposition 7.8.1] and [GL83 , 12-1]. Proposition 2.11. Let G be a simple group of Lie type over Fq and let p be an o* *dd prime not dividing q. Assume a Sylow p-subgroup S of G is abelian and let A = * *1(S). Then NG (A) = NG (S). Proof.The result is trivial if S = A so assume this is not the case; in particu* *lar the exponent of S is at least p2. By part (7) of Proposition 2.9, p does not d* *ivide the order of the Schur multiplier of G, so we may assume G is the (quasisimple) universal cover of the simple group. Clearly NG (S) NG (A). Moreover, since S 2 Sylp(CG (A)), by Frattini's Argument NG (A) = CG (A)NG (S). Thus it suffice* *s to show CG (A) = CG (S). Since CG (A) has an abelian Sylow p-subgroup and since any nontrivial p0-automorphism of S must act nontrivially on A, by Burnside's Theor* *em CG (A) has a normal p-complement. Let = [Op0(CG (A)), S]. It suffices to prov* *e S centralizes . __ Let G be the simply connected universal algebraic group over the algebraic cl* *osure of Fq, and let oe be a Steinberg endomorphism whose fixed points equal G. In t* *he notation of Proposition 2.9, since S = ST, by the proof of [GLSv3 , Theorem 4.1* *0.2] __ __ __ there is a oe-stable maximal torus T of G containing S. Let C denote the connec* *ted __ __ __ __ component of C__G(A ), so C is also oe-stable. Note that T C and since is g* *enerated __ by conjugates of S, so too C . We may now follow the basic ideas in the pro* *of of [GLSv3 , Theorem 7.7.1(d)(2)], where more background is provided. By [SS70 , __ 4.1(b)], C is reductive, so by the general theory of connected reductive groups __ ____ C = Z L __ __ __ where Z is the connected component of the center of C, L is the semisimple comp* *onent __ __ __0 __ (possibly trivial), and Z \ L is a finite group. Since C we have L. The __ group of fixed points of oe on L is a commuting product L1. .L.nof (possibly so* *lvable) groups of Lie type over the same characteristic as G and smaller rank, and S in* *duces inner or diagonal automorphisms on each Li. Since Op0(CG (A)) we have Op0(L1. .L.n) = Op0(L1) . .O.p0(Ln). If Liis a p0-group, then Inndiag(Li) is also a p0-group and so S centralizes Li* *. On the other hand, if p divides the order of Li, then Op0(Li) Z(Li); in this case In* *ndiag(Li) centralizes Z(Li). In all cases S centralizes Op0(Li), as needed. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 15 Proposition 2.12. Let p be any prime, let G be a simple group containing a stro* *ngly closed p-subgroup, let S 2 Sylp(G) and let Z = Z(S). (1) Assume G ~=U3(q) with q = pn, or G ~=Sz(q) with p = 2 and q = 2n. Then S is a special group of type q1+2 or q1+1 respectively, and NG (S) = NG (Z* *) = SH, where the Cartan subgroup H is cyclic of order (q2 - 1)=(3, q + 1) or q * *- 1 respectively. In both families H acts irreducibly on both Z and S=Z, and Z = 1(S) is the unique nontrivial, proper strongly closed subgroup of S. (2) Assume G ~= Re(q) with p = 3 and q = 3n, n > 1. Then S is of class 3, Z ~=Eq and S0= (S) = 1(S) ~=Eq2. Furthermore, NG (S) = NG (Z) = SH, where the Cartan subgroup H is cyclic of order q - 1 and acts irreducibl* *y on all three central series factors: Z and S0=Z and S=S0. Thus Z and 1(S) * *are the only nontrivial proper strongly closed subgroups of S. (3) Assume G ~=G2(q) for some q with (q, 3) = 1 and p = 3. Then Z ~=Z3 is the only nontrivial proper strongly closed subgroup of S. Furthermore, NG (Z* *) ~= fi SLffl3(q) . 2 according to whether 3 fiq - ffl. An element of order 2 in* * NG (Z) - CG (Z) inverts Z, and NG (S)=S ~= QD16 or E4 according as |S| = 33 or |S| > 33 respectively. No automorphism of G of order 3 normalizes S and centralizes both S=Z and a 30-Hall subgroup of NG (S). (4) Assume G is isomorphic to one of the sporadic groups: J2 (with p = 3); Co2, Co3, HS, Mc (with p = 5); or J4 (with p = 11). In each case S is non-abelian of order p3 and exponent p, and Z is the only nontrivial pro* *per strongly closed subgroup of S. The normalizer of Z [in G] is: 3P GL2(9) * *[in J2], 51+2((4 * SL3(3)) . 2) [in Co2], 51+2((4Y S3) . 4) [in Co3], 51+2(8* * . 2) [in HS], (51+2. 3) . 8 [in Mc], or (111+2. SL2(3)) . 10 [in J4]. In G = J2 w* *e have NG (S)=S ~=Z8; and in all other cases NG (S) = NG (A). (5) Assume G ~=J3 with p = 3. Then Z ~=E9 and 1(S) ~=E27 are the only non- trivial proper strongly closed subgroups of S. Furthermore, NG (Z) = NG * *(S) = SH where H ~=Z8 acts fixed point freely on 1(S) and irreducibly on Z. Proof.Part (1) may be found in [HKS72 ] and [Suz62 ]. Part (2) appears in [Wa6* *6 ]. All parts of (4) and (5) appear in [GLSv3 , Chapter 5] with references therein. In part (3), by [GL83 , 14-7] the center of S has order 3 and C = CG (Z) ~=SL* *ffl3(q) fi according to the condition 3 fiq - ffl. The same reference shows G has two conj* *ugacy classes of elements of order 3: the two nontrivial elements of Z are in one cla* *ss, and 16 RAM'ON J. FLORES AND RICHARD M. FOOTE all elements of order 3 in S - Z lie in the other. Now S SLffl3(q) acts abso* *lutely irreducibly on its natural 3-dimensional module over Fq (or Fq2in the unitary c* *ase), hence by Schur's Lemma the centralizer of S in C consists of scalar matrices. T* *hus Z = CC (S) = CG (S). Since the two nontrivial elements of Z are conjugate in G, NG (Z) = C< t > where an involution t may be chosen to normalize S and induce a graph (transpose-inverse) automorphism on C. By canonical forms, all non-centr* *al elements of order 3 in SLffl3(q) are conjugate in GLffl3(q) to the same diagona* *l matrix u = diag(~, ~-1, 1), where ~ is a primitive cube root of unity, but are also co* *njugate in SLffl3(q) to u because the outer (diagonal) automorphism group induced by GL* *ffl3 may be represented by diagonal matrices that commute with u. Thus all elements * *of order 3 in S - Z are conjugate in C. If |S| = 27, then since S=Z is abelian of type (3,3), all elements of order 3* * in S=Z are conjugate under the action of NC (S=Z) = NC (S)=Z; hence they are conjugate under the faithful action of a 30-Hall subgroup, H0, of NC (S) on S=Z. This sh* *ows |H0| 8. Since a 30-Hall subgroup H of NG (S) acts faithfully on S=Z and has o* *rder 2|H0|, it must be isomorphic to a Sylow 2-subgroup, QD16, of GL2(3) as claimed. If |S| = 32a+1> 27 then we may describe S as the group, ST, of diagonal matri* *ces of 3-power order acted upon by a permutation matrix w of order 3 (where < w > = SW* * ). Then ST ~=Z3ax Z3a is the unique abelian subgroup of S of index 3 (as |Z| = 3),* * so NC (S) normalizes ST. Let H0 be a 30-Hall subgroup of NC (S). One easily sees t* *hat H0 must act faithfully on 1(ST) (and centralize Z), hence |H0| 2. Since ther* *e is a permutation matrix of order 2 in C normalizing S, |H0| = 2. Thus NG (S)=S has order 4, and is seen to be a fourgroup by its action on 1(ST). To see that Z is the unique nontrivial strongly closed subgroup that is prope* *r in S suppose B is another, so that Z < B. If B contains an element of order 9 _ henc* *e an element of order 9 represented by a diagonal matrix in C _ then by conjugating * *in C one easily computes that B - Z contains an element of order 3. Since all such* * are conjugate in C this shows 1(S) B. It is an exercise that 1(S) = S (the deta* *ils appear at the end of the proof of Lemma 3.4), a contradiction. Finally, suppose f is an automorphism of G of order 3 that normalizes S and centralizes S=Z. Then |S : CS(f)| 3 so f cannot be a field automorphism as |G2(r3) : G2(r)|3 32 for all r prime to 3. Thus f must induce an inner automo* *r- phism on G, hence act as an element of order 3 in ST. We have already seen that* * no CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 17 such element centralizes a 30-Hall subgroup of NG (S), a contradiction. This co* *mpletes all parts of the proof. Corollary 2.13. Let p be any prime, let L be a finite simple group possessing a strongly closed p-subgroup A that is properly contained in the Sylow p-subgroup* * S of L. Assume further that L is isomorphic to one of the groups Li in the conclusio* *n of Theorem 2.1 or Theorem 2.2. Then one of the following holds: (1) NL(S) = NL(A), (2) |A| = 3 and L ~=G2(q) for some q with (q, 3) = 1, or (3) |A| = 3, L ~=J2 and NL(A) ~=3P GL2(9). Proof.This is immediate from Propositions 2.11 and 2.12. 2.2. The Proof of Theorem 2.2. This subsection derives Theorem 2.2 as a consequence of Theorem 2.3, which is proved in the next section. Throughout this subsection G is a minimal counterex* *ample to Theorem 2.2. Since strong closure inherits to quotient groups, if OA(G) 6= 1 we may apply * *induc- tion to G=OA(G) and see that the asserted conclusion holds. Thus we may assume OA(G) = 1, and consequently A \ N is not a Sylow p-subgroup of N for any nontrivial N E G (2.3) and Op0(G) = 1. Likewise if G0 = < AG > then by Frattini's Argument, G = G0NG (A), whence < AG * *> = < AG0 >. Thus we may replace G by G0 to obtain (2.4) G is generated by the conjugates of A. By strong closure A\Op(G) E G, whence by (2.3), A\Op(G) = 1. Since [A, Op(G)] A \ Op(G) = 1, by (2.4) we have (2.5) Op(G) Z(G). Consequently F *(G) is a product of subnormal quasisimple components L1, . .,.Lr with Op0(Li) = 1 for all i. Moreover Si = S \ Li is a Sylow p-subgroup of Li a* *nd Si6= 1 by (2.3). We argue that each component of G is normal in G. By way of contradiction assume {L1, . .,.Ls} is an orbit of size 2 for the action of G on its compone* *nts. 18 RAM'ON J. FLORES AND RICHARD M. FOOTE Let Z = A \ Z(S), so that Z normalizes each Li. Thus N = \si=1NG (Li) is a proper normal subgroup of G possessing a nontrivial strongly closed p-subgroup, B = A \ N that is not a Sylow subgroup of N. By induction _ keeping in mind that components of N are necessarily components of G and OB (N) = 1 _ and after possible renumbering, there are simple components L1, . .,.Lt of N that s* *atisfy the conclusion of Theorem 2.2 with B \ Li 6= 1, these are all the components of* * N satisfying the latter condition, and t 1. By Frattini's Argument G = NG (B)N * *from which it follows that L1. .L.tE G. The transitive action of G in turn forces t * *= s. Thus A permutes {L1, . .,.Ls} and 1 6= A \ Li< Si. If A does not normalize one * *of these components, say Lai= Lj for some i 6= j and a 2 A, then SiSai= Six Sj. But then [Si, a] 6 (A \ Li) x (A \ Lj), contrary to A E S. Thus A must normalize L* *i for 1 i s. Since A N E G, (2.4) gives N = G, a contradiction. This proves (2.6) every component of G is normal in G. The preceding results also show that A acts nontrivially on each Li. By Lemma 2* *.8, Ai = A \ Li 6= 1 and Ai is not Sylow in Li for every i. By Theorem 2.3 applied * *to each Li using a minimal strongly closed subgroup of Ai we obtain (2.7) F *(G) = L1 x L2 x . .x.Lr and each Li is one of the simple groups described in the conclusion of Theorem * *2.2. Moreover, in each of conclusions (i) to (v), by Propositions 2.10 and 2.12, Ai * *is a subgroup of Li described in the respective conclusion. It remains to verify that the action of A is as claimed when A 6 F *(G). The automorphism group of each Liis described in detail in [GLSv3 , Theorem 2.5.12 * *and Section 5.3] _ these results are used without further citation. Let S* = S \ F *(G) = S1x . .x.Sr, let H* = H1x . .x.Hr, where Hi is a p0-Hall complement to Si in NLi(Si), and let N* = AS*H*. Note that Op0(N*) = CH*(S*) is A-invariant. Now in all cases [A, Si] A \ Si (Si), that is, A commutes * *with the action of H* on S*= (S*). This forces A Op0,p(N*). By strong closure of* * A we get that AOp0(N*) E N*. Thus NN*(A) covers H*=CH*(S*). Let H be a p0-Hall complement to AS* in NN*(A); we may assume H* = HCH*(S*). We have a Fitting decomposition (2.8) A = [A, H]AF where AF = CA(H). CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 19 By Propositions 2.10 and 2.12 each Ai is abelian and Hi, hence also H, acts wit* *hout fixed points on each Ai. Since [A, H] A \ F *(G) we therefore obtain (2.9) [A, H] = A1 x . .x.Ar and AF \ [A, H] = 1. We now determine the action of AF on Li for each isomorphism type in conclusions (i) to (v). First suppose AF acts trivially on some Li, say for i = 1. In this situation* * A = A1x B where B = (A2x . .x.Ar)AF = A \ CG (L1). Then < AG > = L1x < BG >, and so we may proceed inductively to identify < BG > and conclude that Theorem 2.2 * *is valid. We now observe that AF acts trivially on all components listed in conclu* *sions (ii) to (v) as follows: If L1 is one of these cases, it follows from Propositio* *n 2.12 that CH1(S1) = 1 and so AF centralizes a p0-Hall subgroup of NL1(S1). In case (ii) * *of the conclusions, if L1 is a Lie-type simple group in characteristic p and BN-ra* *nk 1, by [GL83 , 9-1] no automorphism of order p centralizes a Cartan subgroup of L1,* * so AF acts trivially on L1. If L1 ~=G2(q) is described by case (iii) of the concl* *usion, then since [S1, AF] A1, the last assertion of Proposition 2.12(3) shows that * *AF acts trivially on L1. And in cases (iv) and (v) of the conclusions, when L1 is a spo* *radic group, none of the target groups admits an outer automorphism of order p, and no inner automorphism that normalizes a Sylow p-subgroup also commutes with a p0-H* *all subgroup of its normalizer. Thus AF acts trivially in these instances too. It remains to consider when every Li is described by conclusion (i): L = Li i* *s a fi group of Lie type over the field Fqiwhere p 6 fiqiand the Sylow p-subgroups are* * abelian but not elementary abelian. Since AF commutes with the action of a p0-Hall subg* *roup of NL(Si), it follows from Proposition 2.10 that AF induces outer automorphisms* * on L. The outer diagonal automorphism group of L has order dividing the order of t* *he Schur multiplier of L, so by Proposition 2.9(7) no element of G induces a nontr* *ivial outer diagonal automorphism of p-power order on L. Since Sylow 3-subgroups of D4(q) and 3D4(q) are non-abelian, L does not admit a nontrivial graph or graph-* *field automorphism when p = 3. This shows AF must act as field automorphisms on L, and hence AF=CAF (L) is cyclic. Now G is generated by the conjugates of A, hence the group eG = G=LCG (L) of outer automorphims on L is generated by conjugates of eAF. This implies via [GL* *Sv3 , Theorem 2.5.12] that (2.10) Ge= eDeAF and eD= [De, eAF] 20 RAM'ON J. FLORES AND RICHARD M. FOOTE where eDis a cyclic p0-subgroup of the outer-diagonal automorphism group of L n* *or- malized by the cyclic p-group eAFof field automorphisms. Moreover, since p > 3 * *when L is of type E6(q), 2E6(q) or D2m(q), the action of eAFon eDin (2.10) implies t* *hat eD is trivial except in the cases where L is a linear or unitary group. A p0-order subgroup D that covers the section eD for every Li may be defined * *as follows (even in the presence of Lithat are not of type (i)): We have now estab* *lished that S = S*AF, and that S* is a Sylow p-subgroup of the (normal) subgroup GD of G inducing only inner and diagonal automorphisms on F *(G). Thus NGD (S*) has a p0-Hall complement, which is then a complement to S = S*AF in NG (S*). Since [S*, AF] (S*), AF commutes with the action on S* of this p0-Hall subgr* *oup. As De = [De, AF], any choice for D must lie in CG (S*). However, CG (S*) has a normal p-complement, so any D must lie in Op0(CG (S*)). Thus [Op0(CG (S*)), AF]* * = [Op0(CG (S*)), S] covers De for every component Li (and centralizes all compone* *nts that are not of type P SL or P SU). Finally note that in every case A0Fcentralizes Li for every i. Since then A0F* *cen- tralizes F *(G), it must be trivial, that is, AF is abelian. Since AF=CAF (Li) * *is cyclic for all i, it follows that AF = AF= \ri=1CAF (Li) has rank at most r, as assert* *ed. This __ completes the proof of Theorem 2.2. |_| 2.3. The Proofs of Corollaries 2.4 and 2.5. Considering both corollaries at once, assume the hypotheses of Corollary 2.4 * *hold. The result is trivial if either A = S (in which case OA(G) = G) or A = 1 (in wh* *ich case G = 1). By passing to G=OA(G) we may assume OA(G) = 1. Since G is generated by conjugates of A, Theorems 2.1 and 2.2 imply that (2.11) G = (L1 x . .x.Lr)(D . AF), where the Li, D and AF are described in their conclusions (with both D and AF trivial when p = 2). Let Si= S \ Li and Ai= A \ Li. For each i let Zi be a minimal nontrivial strongly closed subgroup of A \ Li,* * and let Z = Z1 x . .x.Zr. Then Z is strongly closed in G, and by Propositions 2.10 * *and 2.12, Z is contained in the center of S. It is immediate from Sylow's Theorem a* *nd the weak closure of Z that NG (Z) controls strong G-fusion in S. Now NG (Z) = (NL1(Z1) x . .x.NLr(Zr))(D . AF) CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 21 where by the proof of Theorem 2.2, D = [D, AF] may be chosen to be an S-invaria* *nt p0-subgroup centralizing each Si. This implies (2.12) M = (NL1(Z1) x . .x.NLr(Zr))AF controls strong G-fusion in S. It suffices therefore to show that NM (A) controls strong M-fusion in S. Furthe* *rmore, NM (A) controls strong M-fusion in S if and only if the corresponding fact hold* *s in M=Op0(M); so we may pass to this quotient and therefore assume Op0(M) = 1 (with- out encumbering the proof with overbar notation, since all normalizers consider* *ed are for p-groups). If Li is a Lie-type component with Si abelian then, as noted in the proof of * *Theo- rem 2.2, NLi(Zi) = NLi(Si) and AF commutes with the action on Si of an AF-stable p0-Hall subgroup Hi of this normalizer. Since Op0(M) = 1 it follows that Hi ac* *ts faithfully on Si, and so [AF, Hi] = 1. On the other hand, if Li is not of this* * type, [AF, Li] = 1. Thus (reading modulo Op0(M)) we have (2.13) M = SCM (AF) and so NM (A) = NM (A*), where A* = A1. .A.r. For every component Lithat is not of type G2(q) or J2, by Corollary 2.13, NLi* *(Zi) = NLi(Si); and therefore in these components NLi(Zi) = NLi(Ai) too. However, for a component Li of type G2(q) or J2 (with p = 3), by Proposition 2.12 we must have Zi = Ai. In all cases we have NLi(Zi) = NLi(Ai). Hence NM (A*) = NM (Z) = M and the first assertion of Corollary 2.4 holds by (2.12). This also establishe* *s the second assertion unless p = 3 and some components Liare of type G2(q) or J2, wh* *ere the possibility that |Si| > 33 in these exceptions is excluded by the hypothese* *s of Corollary 2.4. In the remaining case let S* = S1 x . .x.Sr, where S1, . .,.Sk are the Sylow * *3- subgroups of the components of type G2(q) or J2, and Sk+1, . .,.Sr are the rema* *ining ones. Again by (2.13), NM (S) = NM (S*) so we must prove the latter normalizer controls strong M-fusion in S; indeed, it suffices to prove control of fusion i* *n S*. Now NM (S*) controls strong M-fusion in S* if and only if the corresponding res* *ult holds in each direct factor. This is trivial for i > k as Siis normal in that f* *actor. For 1 i k the result is true since Si = 31+2, i.e., Si has a central series 1 <* * Zi < Si whose terms are all weakly closed in Si with respect to NLi(Zi) (see, for examp* *le, [GiSe85 ]). This establishes the final assertion of Corollary 2.4. 22 RAM'ON J. FLORES AND RICHARD M. FOOTE In Corollary 2.5 observe that by Theorem 2.2, once OA(G) is factored out we h* *ave equation (2.11) holding, and since AF acts without fixed points on the cyclic q* *uotient D=(D \ L1. .L.r), we must have NG (A) (L1. .L.r)AF. Thus by (2.13) we have NG (A) = NM (A) SCM (AF)Op0(M). Since NM (A)\Op0(M) centralizes A we have NG (A) SCM (AF), and hence NG (A) = SCM (AF). All parts of Corollary 2.5 now follow. __ |_| 3. The Proof of Theorem 2.3 We now prove Theorem 2.3. Throughout this section p is an odd prime, G is a minimal counterexample, and A is a nontrivial strongly closed subgroup of G tha* *t is a proper subgroup of the Sylow p-subgroup S of G. The minimality implies that if * *H is any proper section of G containing a nontrivial minimal strongly closed (with r* *espect to H) p-subgroup A0, then either A0 is a Sylow subgroup of its normal closure i* *n H ___ __ or the normal closure of A0 in H is a direct product of isomorphic simple grou* *ps, as described in the conclusion of Theorem 2.2, where overbars denote passage to H=OA0(H). In particular, A0 does not even have to be a subgroup of A, although * *for the most part we will be applying this inductive assumption to subgroups A0 A* *\H (which we often show is nontrivial by invoking part (2) of Lemma 2.8). Familiar facts about the families of simple groups, including the sporadic gr* *oups, are often stated without reference. All of these can be found in the excellent,* * ency- clopedic source [GLSv3 ]. Specific references are cited for less familiar resul* *ts that are crucial to our arguments. Lemma 3.1. G is a simple group. Proof.Since strong closure inherits to quotient groups, if OA(G) 6= 1 we may ap* *ply induction to G=OA(G) and see that the asserted conclusion holds. Thus we may assume OA(G) = 1, i.e., (3.1) A \ N is not a Sylow p-subgroup of N for any nontrivial N E G. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 23 In particular, (3.2) Op0(G) = 1. Let G0 = < AG > and assume G0 6= G. By (3.1), A is not a Sylow p-subgroup of G0. Let 1 6= A0 A be a minimal strongly closed subgroup of G0. By the inducti* *ve hypothesis A0 is contained in a semisimple normal subgroup N of G0 satisfying t* *he conclusions of the theorem. Since N E G it follows that M = < NG > is a semisim* *ple normal subgroup of G whose simple components are described by Theorem 2.2. Since A is minimal strongly closed in G and 1 6= A0 A \ M, A M and the conclusion of Theorem 2.3 is seen to hold. Thus (3.3) G is generated by the conjugates of A. By strong closure A \ Op(G) E G, hence by (3.1), A \ Op(G) = 1. Thus [A, Op(G)] A \ Op(G) = 1, i.e., A centralizes Op(G). Since G is generated by conjugates of* * A, (3.4) Op(G) Z(G). By (3.2) and (3.4), F *(G) is a product of commuting quasisimple components, L1* *, . .,.Lr, each of which has a nontrivial Sylow p-subgroup. Since A acts faithfully on F ** *(G), by Lemma 2.8 A \ F *(G) 6= 1. The minimality of A then forces A F *(G). Thus A normalizes each Li, whence so does G by (3.3). Now A acts nontrivially on one component, say L1, so again by Lemma 2.8, A \ L1 6= 1. By minimality of A we obtain A L1 E G, so by (3.3) G = L1 is quasisimple (with center of order a power of p). Finally, assume Z(G) 6= 1 and let eG= G=Z(G). Since A 6= S but A \ Z(G) = 1, by Gasch"utz's Theorem we must have that S 6= AZ(G) and so eAis strongly closed but not Sylow in the simple group eG. Since |Ge| < |G|, the pair (Ge, eA) sati* *sfy the conclusions of Theorem 2.3; in particular, eA= 1(Z(Se)) in all cases. If eGis * *a group of Lie type in conclusion (i), again by Gasch"utz's Theorem together with the irre* *ducible action of NGe(Se) on 1(Se), eAmust lift to a non-abelian group in G. In this s* *ituation Z(G) A0, contrary to A \ Z(G) = 1. In conclusions (ii), (iii) and (iv) the p-* *part of the multipliers of the simple groups are all trivial, so Z(G) = 1 in these case* *s. In case (v) when eG~=J3 and eA= Z(Se) by the fixed point free action of an element of o* *rder 8 in NG (S) on S it again follows easily that A"must lift to the non-abelian gr* *oup of 24 RAM'ON J. FLORES AND RICHARD M. FOOTE order 27 and exponent 3 in G, contrary to A \ Z(G) = 1. This shows Z(G) = 1 and so G is simple. The proof is complete. Lemma 3.2. A is not cyclic and S is non-abelian. Proof.If A is cyclic then since 1(A) is also strongly closed, the minimality o* *f A gives that |A| = p. Then G is not a counterexample by Proposition 2.6. Likewi* *se if S is abelian, by Proposition 2.10 it is homocyclic with NG (S) acting irredu* *cibly and nontrivially on 1(S). By minimality of A we must then have A = 1(S) and the exponent of S is greater than p. None of the sporadic or alternating group* *s or groups of Lie type in characteristic p contain such Sylow p-subgroups, so G must be a group of Lie type in characteristic 6= p. Again, G is not a counterexampl* *e, a contradiction. Note that because A is a noncyclic normal subgroup of S and p is odd, A conta* *ins an abelian subgroup U of type (p, p) with U E S. Furthermore, |S : CS(U)| p so U is contained in an elementary abelian subgroup of S of maximal rank. Lemmas 3.3 to 3.7 now successively eliminate the families of simple groups as possibilities for the minimal counterexample. The argument used to eliminate t* *he alternating groups is a prototype for the more complicated situation of Lie type groups, so slightly more expository detail is included. Lemma 3.3. G is not an alternating group. fi Proof.Assume G ~=An for some n. Since S is non-abelian, n p2. If p 6 fin then* * S is contained in a subgroup isomorphic to An-1, which contradicts the minimality* * of G (no alternating group satisfies the conclusions in Theorem 2.2). Thus n = ps * *for some s 2 N with s p. Let E be a subgroup of S be generated by s commuting p-cycles. Since E contai* *ns a conjugate of every element of order p in G, A \ E 6= 1. We claim E A. Let z = z1. .z.r2 A \ E be a product of commuting p-cycles zi in E with r minimal. * *If r 3 there is an element oe 2 An that inverts both zr and zr-1 and centralizes* * all other zi; and if r = 2, since n 3r there is an element oe 2 An that inverts z* *2 and centralizes z1. In either case, by strong closure zoe2 A \ E and zzoe= z21. .z.* *2r-2or z21 respectively. Hence zzoeis an element of A \ E that is a product of fewer commu* *ting p-cycles, a contradiction. This shows A contains a p-cycle, hence by strong cl* *osure CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 25 E A. Now An contains a subgroup H with (3.5) S H = NAn(E) and H ~=Zp o As. By our inductive assumption H contains a normal subgroup N = OA(H) with E N such that A\N is a Sylow p-subgroup of N and H=N a product of simple components described in Theorem 2.2. Since H is a split extension over E and every element* * of H of order p is conjugate to an element of E, by strong closure A 6= E. Since H=E* * ~=As is not one of the simple groups in Theorem 2.2 it follows that N = H (in the ca* *ses where s = 3 or 4 as well), contrary to A 6= S. This contradiction establishes * *the lemma. Alternatively, one could argue from (3.5) and induction that S = 1(S), and so again S = A by strong closure, a contradiction. Lemma 3.4. G is not a classical group (linear, unitary, symplectic, orthogonal)* * over Fq, where q is a prime power not divisible by p. Proof.Assume G is a classical simple group. Following the notation in [GLSv3 , Theorem 4.10.2], let V be the classical vector space associated to G and let X* * = Isom(V ). We may assume dim V 7 in the orthogonal case because of isomorphisms of lower dimensional orthogonal groups with other classical groups (the dimensi* *on is over Fq2in the unitary case). The tables in [KL90 , Chapter 4] are helpful refe* *rences in this proof. First consider when G is neither a linear group with p dividing q - 1 nor a u* *nitary fi group with p dividing q + 1. This restriction implies that p 6 fi|X : X0| and* * there is a surjective homomorphism X0 ! G whose kernel is a p0-group. Thus we may do calculations in X in place of G (taking care that conjugations are done in X* *0). Proposition 2.9 is realized explicitly in this case as follows: There is a deco* *mposition V = V0 ? V1 ? . .?.Vs of V (? denotes direct sum in the linear case), where Isom(V0) is a p0-group, t* *he cyclic group of order p has an orthogonally indecomposable representation on each othe* *r Vi, the Vi are all isometric, and a Sylow p-subgroup of Isom(Vi) is cyclic. Further* *more, X0 contains a subgroup isomorphic to As permuting V1, . .,.Vs and the stabilize* *r in X0 of the set {V1, . .,.Vs} contains a Sylow p-subgroup of X. In other words, w* *e may 26 RAM'ON J. FLORES AND RICHARD M. FOOTE assume (3.6) S H ~=Isom (V1) o As. In the notation of Proposition 2.9, let S \ Isom(Vi) = < ui>, where uiacts triv* *ially on Vj for all j 6= i. Then ST = < u1, . .,.us> and SW is a Sylow p-subgroup of As* *. Since S is non-abelian, SW 6= 1 and so s p 3. Let zi be an element of order p in* * < ui>, and let E = < z1, . .,.zs> = 1(ST) ~=Eps. The faithful action of SW on ST forces Z(S) ST, so A \ E 6= 1. We claim E A. As in the alternating group case, let z be a nontrivial eleme* *nt in A \ E belonging to the span of r of the basis elements zi in E with r minima* *l. After renumbering and replacing each zi by another generator for < zi> if neces* *sary, we may assume z = z1. .z.r. If r 3 there is an element oe 2 G that acts trivi* *ally on z1, . .,.zr-2 and normalizes but does not centralize < zr-1, zr>; and if r =* * 2, since s 3 there is an element oe 2 G that centralizes z1 and normalizes but does not centralize < z2>. In both cases zoez-1 is a nontrivial element of A \ E that is* * a product of fewer basis elements. This shows zi 2 A for some i and so E A since all zj* * are conjugate in G. By Proposition 2.9(5) in this setting, every element of order p in G is conju* *gate to an element of E. Since the extension in (3.6) is split, A 6 ST. By the ov* *erall induction hypothesis applied in H (or because a Sylow p-subgroup of As is gener* *ated by elements of order p), it follows that A covers S=ST. We may therefore choose* * a numbering so that for some x 2 A, ux1= u2. Thus u = u1u-12= [u2, x] 2 A \ Isom(V1 ? . .?.Vs-1). Let Y = G \ Isom(V1 ? . .?.Vs-1) so that Y is also a classical group of the same type as G over Fq. Note that the dimension of the underlying space on which Y acts is at least 2(s - 1) by our initial restrictions on q. Since Y is proper * *in G, by induction applied using a minimal strongly closed subgroup A0 of A \ Y in Y we obtain the following: either A0 (hence also A) contains a Sylow p-subgroup of Y* * , or the Sylow p-subgroups of Y are homocyclic abelian with A0 \ Y elementary abelian of the same rank as a Sylow p-subgroup of Y . Furthermore, in the latter case a Sylow p-normalizer acts irreducibly on A0, and hence the strongly closed subgro* *up A \ Y is also homocyclic abelian. Since A \ Y contains the element u of orde* *r d, CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 27 where d = |u1|, in either case A \ Y contains all elements of order d in S \ Y * *. Since u1 2 S \ Y this proves u1 2 A. By (3.6) all uiare conjugate in G to u1, hence S* *T A and so A = S a contradiction. It remains to consider the cases where V is of linear or unitary type and p d* *ivides fi q - 1 or q + 1 respectively (denoted as usual by p fiq - ffl). Now replace the * *simple group G by its universal quasisimple covering SLffl(V ). Likewise replace A by* * the p-part of its preimage. Thus A is a noncyclic (hence noncentral) strongly close* *d p- subgroup of SLffl(V ). In this situation S = STSW where we may assume ST is t* *he group of p-power order diagonal matrices of determinant 1 (over Fq2 in the unit* *ary case), and SW is a Sylow p-subgroup of the Weyl group W of permutation matrices permuting the diagonal entries. Furthermore, ST is homocyclic of exponent d, wh* *ere d = |q - ffl|p, and is a trace 0 submodule of the natural permutation module fo* *r W of exponent d and rank m = dim V . Since A is noncyclic, it contains a noncentr* *al element z of order p; and by Proposition 2.9, z is conjugate to an element of S* *T, i.e., is diagonalizable. Arguing as above with E = 1(ST) we reduce to the case where* * z is represented by the matrix diag(i, i-1, 1, . .,.1) for some primitive pthroot* * of unity i. The action of W again forces E A. Again, every element of order p in S is conjugate in G to an element of E, so by strong closure (3.7) 1(S) A. Consider first when m 5. Then CG (z) contains a quasisimple component L ~= SLfflm-2(q)0. Since L contains a conjugate of z, the inductive argument used i* *n the general case shows that A \ L contains a diagonal matrix element of order d, he* *nce contains such an element centralizing an n - 2 dimensional subspace. The strong closure of A then again yields ST A; and as before by induction or because S = ST 1(S) we get A = S, a contradiction. Thus dim V 4, and since SW 6= 1 we must have p = 3. If G ~=SLffl4(q) then * *let z be represented by the diagonal matrix diag(i, i, i, 1), where i is a primitive * *3rdroot of unity. Then CG (z) contains a Sylow 3-subgroup of G and a component of type SLffl3(q), so the preceding argument leads to a contradiction. Finally, consider when G ~=SLffl3(q). The Sylow 3-subgroups of SLffl3(q) are * *described in the proof of Proposition 2.12. In both instances ST is homocyclic of rank 2* * and exponent d with generators u1, u2, and with SW = < w > ~=Z3 acting by uw1= u2 and uw2= u-11u-12. 28 RAM'ON J. FLORES AND RICHARD M. FOOTE Thus u1w has order 3, and so u1 = (u1w)w-1 2 1(S). By (3.7), this again forces A = S, which gives the final contradiction. Lemma 3.5. G is not an exceptional group of Lie type (twisted or untwisted) over Fq, where q is a prime power not divisible by p. fi Proof.Assume G = L(q) is an exceptional group of Lie type over Fq with p 6 fiq. Throughout this proof we rely on the Sylow structure for G as described in Prop* *o- sition 2.9. It shows, in particular, that we need only consider when the odd pr* *ime p divides both order of the Weyl group of the untwisted group corresponding to G * *and fi pm0 fim for some m 2 O(G); in all other cases the proposition gives that the Sy* *low p-subgroup is homocyclic abelian. The cyclotomic factors m (q) and their "mult* *iplic- ities" rm for each of the exceptional groups are listed explicitly in [GL83 , T* *able 10:2]. fi fi Note that 3 fiq2 - 1, so in this case m0 is 1 or 2; also, 5 fiq4 - 1, so in thi* *s case m0 fi is 1, 2, or 4; finally, 7 fiq6 - 1, so in this case m0 is 1, 2, 3, or 6. In the* * notation of Proposition 2.9, except in the case 3D4(q) we have S = STSW (split extension) * *where ST is a normal homocyclic abelian subgroup of exponent | m0(q)|p and rank rm0, * *and |SW | = pb, where b is defined in (2.2). The exceptional groups are listed in Table 3A along with p dividing the order* * of the Weyl group, permissible m0 such that m = pam0 for some m 2 O(G) with a 1, and the corresponding rm0 and pb for each of these (in the case of 3D4(q) we de* *fine 3b so that |S| = (| m0(q)|p)rm03b). We consider all these cases, working from largest to smallest _ the latter re* *quiring more delicate examination. Table 4-1 in [GL83 ] is used frequently without spe* *cific citation: it lists all the "large" subgroups of various families of Lie type gr* *oups that we shall employ. It is helpful to keep in mind the description of the order of * *a Sylow p-subgroup in Proposition 2.9 when comparing the p-part of |G| to that of its L* *ie-type subgroups. Case pp==77p:=E78(q) contains both A8(q) and 2A8(q) and so, by inspection of or* *ders, shares a Sylow 7-subgroup with it in the cases (1,8,7) and (2,8,7) respectively* * (the Sylow 7-subgroup order is seen to be 7 . |q - ffl|87for each group). Likewise * *E7(q) contains both A7(q) and 2A7(q) and so shares a Sylow 7-subgroup with it in the * *cases (1,7,7) and (2,7,7) respectively. By minimality of G all the p = 7 cases are el* *iminated. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 29 Table 3A ____Group______Prime_p_____Permissible_(m0,_rm0,_pb)______ 3D4(q) 3 (1, 2, 32), (2, 2, 32) G2(q) 3 (1, 2, 3), (2, 2, 3) F4(q) 3 (1, 4, 32), (2, 4, 32) 2F4(2n)0 3 (2, 2, 3) E6(q) 3 (1, 6, 34), (2, 4, 32) 5 (1, 6, 5) 2E6(q) 3 (1, 4, 32), (2, 6, 34) 5 (2, 6, 5) E7(q) 3 (1, 7, 34), (2, 7, 34) 5 (1, 7, 5), (2, 7, 5) 7 (1, 7, 7), (2, 7, 7) E8(q) 3 (1, 8, 35), (2, 8, 35) 5 (1, 8, 52), (2, 8, 52), (4, 4, 5) 7 (1, 8, 7), (2, 8, 7) Case pp==55p:=T5he same containments in the preceding paragraph for E7(q) show * *these groups share a Sylow 5-subgroup in cases (1,7,5) and (2,7,5). Similarly, E8(q) * *contains SU5(q2) and shares a Sylow 5-subgroup with it in the case (4,4,5). By minimali* *ty these p = 5 cases are eliminated. Assume G ~= E8(q). Using the same large subgroups as in the p = 7 case, the Sylow 5-subgroup S has a subgroup S0 of index 5 that lies in a subgroup G0 of G of type A8(q) or 2A8(q) according to whether we are in cases (1, 8, 52) or (2, * *8, 52) respectively. By Proposition 2.9 applied to G0 it follows that S0 is non-abelia* *n; and since |A| > 5, A \ S0 6= 1. Thus by induction applied to a minimal strongly clo* *sed subgroup A0 A \ S0 in G0 we obtain S0 A. Moreover, by Proposition 2.9 it follows that ST S0. Since A is non-abelian and since the normalizer of a Syl* *ow 5-subgroup of the Weyl group of E8 acts irreducibly on the Sylow 5-subgroup of W (which is abelian of type (5,5)), the strongly closed subgroup A containing ST * *cannot have index 5 in S, a contradiction. This eliminates all E8(q) cases for p = 5. 30 RAM'ON J. FLORES AND RICHARD M. FOOTE * * fi Adopting the notation following Proposition 2.9, assume G ~=Effl6(q), where 5* * fiq -ffl and ST has rank 6 and index 5 in S. Then G shares the Sylow 5-subgroup S with G0 = L1*L2, where L1 and L2 are central quotients of SLffl2(q) and SLffl6(q) re* *spectively (both of whose centers have order prime to 5). Since A is not cyclic, it does * *not centralize L2; hence it follows from Lemma 2.8 that A \ L2 6= 1. Since S \ L2 * *is non-abelian, by induction S \ L2 A. In particular, A contains a homocyclic ab* *elian subgroup of rank 5 and exponent |q - ffl|5, and S=A is cyclic. Now G also cont* *ains a subgroup G1 = K1 * K2 * K3 with each Ki ~= SLffl3(q), where we may assume S \ G1 2 Syl5(G1). Each Ki contains a homocyclic abelian subgroup Bi of rank 2 and exponent |q - ffl|5 with NKi(Bi) acting irreducibly on 1(Bi). Because S=A* * is cyclic it follows that B1 x B2 x B3 = ST A; and since A is non-abelian, A = S. This completes the elimination of all p = 5 cases. We next consider the various p = 3 cases, leaving the nettlesome groups of ty* *pe G2(q) and 3D4(q) until the very end. fi Case pp==33pa=n3d m0m=01=m10=:1Here 3 fiq - 1. If G ~=F4(q) then it contains th* *e universal group G0 = B4(q)u. By inspection of the order formulas, G0 may be chosen to con* *tain a subgroup S0 of index 3 in S which, by Proposition 2.9, is non-abelian. Since * *|A| > 3 we have S0 \ A 6= 1 so, as usual, the minimality of G forces S0 A. Thus S0 = A has index 3 in S. Furthermore, since a Sylow 3-subgroup of the Weyl group of B4 has order 3, we get that A has an abelian subgroup of index 3. But now by [GLSv* *3 , 0 Table 4.7.3A] there is an element t of order 3 in G such that C = O3 (CG (t)) =* * L1*L2 where Li~= SL3(q) for i = 1, 2. Choose a suitable representative of this class * *so that CS(t) 2 Syl3(C). Then A \ Li 6 Z(Li), so because each Sylow subgroup S \ Li is non-abelian, by induction S \ Li A for i = 1, 2. This gives a contradiction be* *cause S \ L1L2 clearly does not have an abelian subgroup of index 3. Since 2E6(q) shares a Sylow 3-subgroup with a subgroup of type F4(q) this fam* *ily is eliminated by minimality of G. Consider when G is one of E6(q), E7(q) or E8(q). In these cases ST is homocyc* *lic of the same rank as G and ST lies in a maximal split torus T of G with W = NG (T )=CG (T ) isomorphic to the Weyl group of G. Note that W acts on the Sylo* *w 3- subgroup ST of T ; moreover, in each case W acts irreducibly on 1(ST), and Z(S) ST. By strong closure of A we obtain (3.8) 1(ST) A. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 31 There are containments: F4(q) E6(q) E7(q) E8(q), with corresponding con- tainments of their maximal split tori. Thus by (3.8), in each exceptional fami* *ly A nontrivially intersects a subgroup, G0, of G of smaller rank in this chain. Sin* *ce the Sylow 3-subgroups of each G0 are non-abelian, by minimality of G and the preced* *ing results we get that A contains a Sylow 3-subgroup of the respective subgroup G0. Since then A is non-abelian, it is not contained in ST. Now the Weyl group of G* * is of type U4(2) . 2, Z2 x S6(2), or 2 . O+8(2) . 2, so by induction applied in NG* * (T ) it follows that A covers a Sylow 3-subgroup of W . Finally, the irreducible action* * of W on ST= (ST) forces ST A, and so A = S, a contradiction. fi fi Case pp==33pa=n3d m0m=02=m20=:2Here 3 fiq + 1. The argument employed when 3 fi* *q - 1 mutatis mutandis eliminates F4(q) as a possibility (using Li ~=SU3(q) in this c* *ase). The groups 2F4(2n)0_ including the Tits simple group _ share a Sylow 3-subgroup with their subgroups SU3(2n), and so are eliminated by induction. Also, E6(q) s* *hares a Sylow 3-subgroup with its subgroup F4(q), hence it is eliminated. To elimina* *te E8(q), E7(q) and 2E6(q) we refer to the table of centralizers of elements of or* *der 3 in these groups: [GLSv3 , Table 4.7.3A]. First assume G ~=E8(q). By [GLSv3 , Table 4.7.3B], G contains a subgroup X ~= L1x L2, where the two components are conjugate and of type U5(q). We may assume S \ X 2 Syl3(X). Since 1(ST) is the unique elementary abelian subgroup of S of rank 8, 1(ST) X; in particular, A \ X 6= 1. As usual, by minimality of G we obtain S \X A, and the "toral subgroup" for S \X lies in ST. Order considerat* *ions then give ST A and |S : A| 33. Now the centralizer of an element of order 3* * in Z(S) is of type (2E6(q) * SU3(q))3, where the two factors share a common center* * of order 3. Since ST A it follows that A acts nontrivially on, hence contains a * *Sylow 3-subgroup of, each component (or of SU3(2) when q = 2). This implies A covers S=ST ~=SW , as needed to give the contradiction A = S. Let G ~=E7(q). Then G contains a subgroup X ~=SU8(q) with S \ X 2 Syl3(X). Since S \ X has the same "toral subgroup" as S, as usual we obtain S \ X A, ST A and |S : A| 32. Now S also contains an element of order 3 whose centralizer has a component of type 2E6(q) (universal version). Since as usual* * A contains a Sylow 3-subgroup of this component it follows that A covers S=ST and* * so A = S, a contradiction. 32 RAM'ON J. FLORES AND RICHARD M. FOOTE Finally, assume G ~=2E6(q). Since by [CCNPW ] 2E6(2) shares a Sylow 3-subgr* *oup with a subgroup of type F i22, by minimality of G we may assume q > 2. Let X be the centralizer of an element of order 3 in Z(S), so X ~= (L1 * L2 * L3)(3 x* * 3), where each Li ~=SU3(q), the central product L1L2L3 has a center of order 3, an element of S cycles the three components, and another element of S induces outer diagonal automorphisms on each Li. As usual, it follows easily that A contains* * a Sylow 3-subgroup of S \ X. By order considerations |ST : ST \ A| 3 and |S : A| 9. Now there is an element t of order 3 in S such that C = CG (t) = D * T1, where D ~=D-5(q) and T1 ~=Zq+1, and we may choose t so that S0 = CS(t) 2 Syl3(C). Let S1 = S \ D and S2 = S \ T* *1, and note that < t > = 1(T1). Since the Schur multiplier of D has order prime t* *o 3, S0 = S1 x S2. It follows as usual that S1 A. Now let w 2 S - S0 and let t1 = tw. Then t1 6= t and S0 2 Syl3(CG (t1)). By symmetry, the strongly closed subgroup A contains the Sylow 3-subgroup Sw1of the component Dw of CG (t1). Since t1 acts faithfully on D, so too Sw2acts faithful* *ly on D, from which it follows that S2 S1Sw1 A. Moreover, A contains the "toral subgroup" of C of type (q + 1)6 (in the univers* *al version of G), so ST A and hence A is the subgroup of S that normalizes each component Li of X. Since SW is generated by elements of order 3 (in the univer* *sal version of G), S = A< x > for some element x of order 3. Since no conjugate of * *x lies in A we may further assume CS(x) 2 Syl3(CG (x)). Since < x > cycles L1, L2, L2 it * *follows that the 3-rank of CG (x) is at most 5: this restricts the possibilities for th* *e type of x in [GLSv3 , Table 4.7.3A]. In all possible cases CG (x) contains a product, L* *, of one or two components with C(L) cyclic. The same argument that showed S2 A may now be applied to show x 2 A, a contradiction. This completes the proof for the* *se families. Case G2(q)G2(q)G2(q)and 3D4(q)3D4(q)3D4(q)where qqq fflf(modfl3)(mod: 3)fflI(mo* *df 3)G ~= G2(q) then by Proposi- tion 2.12 Z(S) ~=Z3 is the unique candidate for A, contrary to Lemma 3.2. Thus * *the minimal counterexample is not of type G2(q). CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 33 Assume G ~= 3D4(q). Then G contains a subgroup G0 isomorphic to G2(q) (the fixed points of a graph automorphism of order 3), and by order considerations we may assume S0 = S \ G0 is Sylow in G0 and so has index 3 in S. As noted above, < z > = Z(S0) is of order 3 and is the unique nontrivial strongly closed (in G0* *) proper subgroup of S0. Consider first when |A \ S0| > 3. Then since S0 is non-abelia* *n, induction applied to G0 gives S0 A, and so A = S0. Since by Proposition 2.6, zG0\S0 = {z 1}, whereas < z > is not strongly closed in G, there must be G-conj* *ugates of z in S-S0, contrary to A being strongly closed (one can see this fusion in a* * subgroup of 3D4(q) of type P GLffl3(q)). Thus A \ S0 = < z > and so by Lemma 3.2, A = < z > x < y > with z ~ y in G. Since [S, y] < z >, y centralizes (S). Since 3D4(q) has 3-rank 2 and y =2 (S* *), by Proposition 2.9(4) we must have |S| = 34. But then S0 is the non-abelian group * *of order 27 and exponent 3, and y centralizes a subgroup of index 3 in it, contrar* *y to the 3-rank of 3D4(q) being 2. This eliminates the possibility that G ~=3D4(q) a* *nd so completes the consideration of all cases. Lemma 3.6. G is not a group of Lie type (untwisted or twisted) in characteristi* *c p. Proof.Assume G is of Lie type (untwisted or twisted) over Fq where q = pn. Sinc* *e G is a counterexample, it follows from Proposition 2.10 that G has BN-rank 2. An end-node maximal parabolic subgroup P1 for each of the Chevalley groups (untwis* *ted or twisted) containing the Borel subgroup S is described in detail in [CKS76 ]* * and [GLS93 ] (for the classical groups these parabolics are the stabilizers in G of* * a totally isotropic one-dimensional subspace of the natural module.) For the groups of B* *N- rank 2 the other maximal parabolic, P2, is also described in [GLS93 ]. In each * *group Pi= QiLiH, where Qi= Op(Pi), Li is the component of a Levi factor of Pi and H is a p0-order Cartan subgroup. Except for the 5-dimensional unitary groups and some groups over F3 (which wi* *ll 0 be dealt with separately), for some i 2 {1, 2} the group M = Op (Pi) satisfies * *the following conditions: Properties 3A. (1) S M, (2) F *(M) = Op(M), ___ (3) M = M=Op(M) is a quasisimple group of Lie type in characteristic p, ___ (4) M is not isomorphic to U3(pn) or Re(3n) (when p = 3), for any n 2, 34 RAM'ON J. FLORES AND RICHARD M. FOOTE ___ (5) [Op(M), M ] = Op(M), and (6) if Q = Op(M) and Z = 1(Z(S)), then one of the following holds: (i):Q is elementary abelian of order qk for some k, or (ii):Q is special of type q1+k for some k, all subgroups of order p in* * Z are conjugate in G, and zg 2 S - Q for some z 2 Z, g 2 G. Basic information about this parabolic is listed in Table 3B. The last column* * of Table 3B indicates which of the two alternatives in Properties 3A(6) holds. The* * proofs that the fusion in Properties 3A(6ii) holds in each case may be found in [CKS76* * ]. Table 3B ____Group________________Parabolic________Q_________L=Z(L)______3A(6)_____ Lk(q), k 3 P1 qk-1 Lk-1(q) (i) Ok (q), k 7 P1 qk-2 Ok-2(q) (i) S2k(q), k 2 P1 q1+2(k-1) S2k-2(q) (ii) Uk(q), k 4, k 6= 5 P1 q1+2(k-2) Uk-2(q) (ii) E6(q) P1 q1+20 L6(q) (ii) E7(q) P1 q1+32 O+12(q) (ii) E8(q) P1 q1+56 E7(q) (ii) 2E6(q) P1 q1+20 U6(q) (ii) G2(q), q > 3 P2 q1+4 L2(q) (ii) F4(q) P1 q1+14 S6(q) (ii) 3D4(q) P2 q1+8 L2(q3) (ii) U5(q) P1 q1+6 U3(q) (ii) 0 Putting aside the last row for the moment, let M = Op (Pi) be chosen according to Table 3B. Since M does not have any composition factors isomorphic to U3(pn)* * or Re(3n), the minimality of G gives inductively that A 2 Sylp(< AM >). If A 6 Q,* * then by the structure of M in Properties 3A(3) and (5), M < AM >. But then A = S by (1), a contradiction. Thus (3.9) A Q and A E M. Assume first that Properties 3A(6ii) holds. Then since A E S, Z \ A 6= 1. T* *he strong closure of A together with (6ii) forces Z A, contrary to the existence* * of CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 35 some zg 2 S - Q. This contradiction shows that G can only be among the families in the first two rows or the last row of Table 3B. Assume now that Q is abelian, i.e., G is a linear or orthogonal group. In th* *ese ___ * *___ cases Q is elementary abelian and is the natural module for M ; in particular, M acts irreducibly on Q. By (3.9) we obtain A = Q. However, in these cases when G is viewed as acting on its natural module, Q is a subgroup of G that stabilizes* * the one-dimensional subspace generated by an isotropic vector and acts trivially on* * the quotient space. Since the dimension of the space is at least 3, one easily exh* *ibits noncommuting transvections that stabilize a common maximal flag; hence there are conjugates of elements of Q in S that lie outside of Q, a contradiction. In U5(q) for q 3 the unipotent radical of the parabolic P1 is special of ty* *pe q1+6 with Z = Z(S) = Z(Q1) and all subgroups of order p in Z conjugate in P1 (so Z * * A). As in the other unitary groups, there exist z 2 Z and g 2 G such that zg 2 S - * *Q1. Now L1 ~=U3(q) acts irreducibly on Q1=Z and, by the strong closure of A, A \ Q * *is normal in P1. Since zg 2 A and [Q1, zg] A \ Q1, the irreducible action of L1 * *forces Q1 A. But now there is a root group U of type U3(q) with U contained in Q1 su* *ch that S = Q1Ux, for some x 2 G. Since U A, this forces A = S, a contradiction. It remains to treat the special cases when the Levi factors in Table 3B are n* *ot quasisimple: G ~=L2(q), L3(3), G2(3), S4(3), or U4(q) (in line 3 of Table 3B, S* *2(q) = L2(q)). Properties of small order groups may be found in [CCNPW ]. The grou* *ps L2(q) have elementary abelian Sylow p-subgroups so G is not a counterexample in this instance. In L3(3) we have S ~=31+2 and the action of the two maximal para* *bolic subgroups (stabilizers of one- and two-dimensional subspaces) easily show that * *the strong closure Z(S) in S is all of S, contrary to A 6= S. If G ~=G2(3) then since G has two (isomorphic) maximal parabolics containing * *S, A is not normal in one of them, say P1. By [CCNPW ], P1 = (W x U) : L where W ~= 31+2, U ~=Z3 x Z3, O3(P1) = W U, and L ~=GL2(3) acts naturally on both U and W=W 0. Since A projects onto a subgroup of order 3 in P1=O3(P1) ~=L, we see that [A, W ] 6 W 0and [A, U] 6= 1. Both these commutators lie in the strongly * *closed subgroup A, so the action of L forces O3(P ) A. Thus A = S, a contradiction. If G ~=S4(3) there are maximal parabolics of type P1 = 31+2 : SL2(3) and P2 = 33(S4 x Z2). Since P1 = NG (Z(S)) it follows that the S4 Levi factor in P2 acts irreducibly on O3(P2). Now A \ O3(P2) 6= 1 so O3(P2) A. Likewise since A is* * a 36 RAM'ON J. FLORES AND RICHARD M. FOOTE noncyclic strongly closed subgroup, it follows easily from the action of the Le* *vi factor in P1 that O3(P1) A. These together give A = S, a contradiction. Finally, assume G ~= U4(q). From the isomorphism U4(q) ~= O+6(q) we see that G contains a maximal parabolic P2 = q4O+4(q) ~= q4L2(q2), where the Levi factor is irreducible on the (elementary abelian) unipotent radical. This case has be* *en eliminated by previous considerations. This final contradiction completes the p* *roof of the lemma. Lemma 3.7. G is not one of the sporadic simple groups. Proof.The requisite properties of the sporadic groups for this proof are nicely* * doc- umented in [CCNPW ], [GL83 , Section 5], or [GLSv3 , Section 5.3]; many of th* *eir proofs may be found in [Asc94 ]. Facts from these sources are quoted without f* *ur- ther attribution. Verification that the sporadic groups in conclusions (iv) and* * (v) of Theorem 2.2 indeed have strongly closed subgroups as asserted may also be found* * in these references. We clearly only need to consider groups where p2 divides the * *order; indeed, when the Sylow p-subgroup has order exactly p2 it is elementary abelian* * and G is not a counterexample in these cases. If |S| = p3, then in all cases the Sylow p-subgroup is non-abelian of exponen* *t p and, with the exception of M12, NG (S) acts irreducibly on S=Z(S). In M12 with p = 3: S contains distinct subgroups U1 and U2, each of order 9, such that NG (* *Ui) acts irreducibly on Uifor each i. Since A is noncyclic and strongly closed, in * *all cases these conditions force A = S, a contradiction. Thus we are reduced to consider* *ing when |S| p4. We first argue that the following general configuration cannot occur in G: Properties 3B. (1) Z(S) = Z ~=Zp, (2) N = NG (Z) has Q = Op(N) extraspecial of exponent p and width w > 1 (denoted Q ~=p1+2w), (3) N acts irreducibly on Q=Q0, and (4) N=Q does not have a nontrivial strongly closed p-subgroup that is proper* * in a Sylow p-subgroup of N=Q. By way of contradiction assume these conditions are satisfied in G. If A 6 Q then by (4) we obtain that A covers a Sylow p-subgroup of N=Q. In this case, the CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 37 irreducible action of N on Q=Q0 then forces Q A and so A = S, a contradiction. Thus A Q. Now Z A but |A| > p so the irreducible action of N forces A = Q. Since A is minimal strongly closed, whence Z is not strongly closed, there is s* *ome x 2 Q - Z such that x ~ z for z 2 Z. Thus by Sylow's Theorem there is some g 2 G such that CQ (x)g S and xg = z. By strong closure, CQ (x)g Q. But since Q has width > 1 we obtain Zg -1 (CQ (x)g)0 Q0= Z and so g normalizes Z. This contradicts the fact that zg =2Z and so proves these properties cannot hold in G. Most sporadic groups are eliminated because they satisfy Properties 3B, or be* *cause they share a Sylow p-subgroup with a group that is eliminated inductively. All * *cases where |S| p4 are listed in Table 3C along with the isomorphism type of the co* *rre- sponding normalizer of a p-central subgroup (or another "large" subgroup, or re* *ason for elimination). Some additional arguments must be made in a few cases. When p = 5 and G ~=Co1 the extraspecial Q = O5(N) listed in the table has wid* *th 1. As before, if A 6 Q then the irreducible action of N on Q=Q0 forces A = S,* * a contradiction. Thus A Q and again the irreducible action yields A = Q. However G contains a subgroup G0 ~=Co2 whose Sylow 5-subgroup S0 is isomorphic to Q and has index 5 in S. Since |A \ S0| 25, the irreducible action of NG0(S0) on S0* *=S00 forces S0 A, and hence S0 = A. But by Proposition 2.6, Z(S0) is strongly clos* *ed in G0 but not strongly closed in G. Thus there is some g 2 G such that Z(S0)g * * S but Z(S0)g 6 S0. This contradicts the fact that A = S0 is strongly closed in G* *, and so G 6~=Co1. When p = 3 and G ~=F i23 it contains a subgroup H of type O+8(3) : S3 that may therefore be chosen to contain S. Let H0 = H00~=O+8(3). By Lemma 2.8, A\H0 6= 1; and so by induction A contains the non-abelian Sylow 3-subgroup S0 = S \ H0 of H0. Thus |S : A| = 3. Now H is generated by 3-transpositions in G, and so there are 3-transpositions t, t1 such that D1 = < t, t1> ~=S3 and H = H0 : D1. Likewise t inverts some element of order 3 in H0, i.e., there is some t2 2 H0< * *t > such that D2 = < t, t2> ~=S3. By the rank 3 action of G on its 3-transpositions, D1 * *and D2 38 RAM'ON J. FLORES AND RICHARD M. FOOTE Table 3C ___Group______Z(S)__normalizer_(or_other_reason)_______ pp==77p = 7 M 71+4(3 x 2S7) pp==55p = 5 Ly 51+4((4 * SL2(9)).2) Co1 51+2GL2(5) HN 51+4(21+4(5 . 4)) B 51+4(((Q8 * D8)A5) . 4) M 51+6((4 * 2J2) . 2) pp==33p = 3 McL 31+4(2S5) Suz 3U4(3)2 Ly 3McL2 O'N (one class of Z3 and S = 1(S)) Co1 31+4GSp4(3) Co2 31+4((D8 * Q8) . S5) Co3 31+4((4 * SL2(9)) . 2) F i22 (S O7(3)) F i23 (S O+8(3) : S3) F i024 31+10(U5(2) . 2) HN 31+4(4 * SL2(5)) T h (see separate argument) B 31+8(21+6O-6(2)) M 31+12(2Suz) . 2 are conjugate in G. Thus D01is conjugate to the subgroup D02of H0, contrary to A being strongly closed. This proves G 6~=F i23. Finally, assume p = 3 and G ~= T h. Following the Atlas notation and the com- putations in [Wi98 ], the centralizer of an element of type 3A in S has isomorp* *hism CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 39 type N = NG (< 3A >) ~=(Z3 x H).2 where H ~=G2(3). Since an element of type 3B in Z(S) \ A commutes with 3A and therefore acts nontrivially on H, by induction A contains a Sylow 3-subgroup of H. In the Atl* *as notation for characters of G2(3), the character O of degree 248 of T h restrict* *s to Z3 x H as O|Z3xH = 1 (O1 + O6) + (! + __!) O5 where the characters of the Z3 factor are denoted by their values on a generato* *r. By comparison of the values of these on the G2(3)-classes it follows that H contai* *ns a representative of every class of elements of order 3 in T h. The calculations i* *n [Wi98 ] show that S = 1(S), which leads to A = S, a contradiction. This eliminates all sporadic simple groups as potential counterexamples, and * *so completes the proof of Theorem 2.3. 4. The BZ=p-cellularization of classifying spaces of finite groups Throughout this section p is an arbitrary prime, G is a finite group and S 2 Sylp(G). Now that we have described precisely the structure of the finite grou* *ps possessing a strongly closed p-subgroup, we are prepared to analyze in detail h* *ow this is related with the BZ=p-cellular structure of the classifying spaces of t* *he groups. Before undertaking the complete description of CW BZ=pBG we describe what it is known so far about this problem. 4.1. Previous results. As we said in the introduction, the starting point was t* *he computation done by Dror-Farjoun in [Far95, 3.C], where he establishes that the BZ=p-cellularization of the classifying space of a finite cyclic p-group has th* *e homo- topy type of BZ=p. Subsequently Rodr'iguez-Scherer investigated in [RS01 ] the M(Z=p, 1)-cellula* *rization, where M(Z=p, 1) denotes the corresponding Moore space for Z=p. When the target is BG, this can be considered a precursor to our study because M(Z=p, 1) can be described as the 2-skeleton of BZ=p. In their description the authors use the c* *oncept of cellularization in the category of groups (developed afterwards in [FGS07 ])* *. Their work in this subject allows one to prove, in particular, that the BZ=p-cellular* *ization of the classifying space of a p-group is the same as that of its p-socle; as th* *e latter is 40 RAM'ON J. FLORES AND RICHARD M. FOOTE BZ=p-cellular in this case ([Flo07, Proposition 4.14]), one obtains that CW BZ* *=pBG ' B 1(G) if G is a finite p-group. The aforementioned is proved using a characterization of the cellularization * *discov- ered by Chach'olski, that is perhaps the most useful tool available to attack t* *hese kind of problems. Because of its importance and ubiquity in our context we reproduce* * it here: Theorem ([Cha96 , 20.3]). Let A and X be pointed spaces, and let C be the homot* *opy W cofiber of the map [A,X]*A ! X, defined as evaluation over all the homotopy c* *lasses of maps A ! X. Then CW BZ=pX has the homotopy type of the homotopy fiber of the composition X ! C ! P A C. Here P denotes the nullification functor, first defined by A.K. Bousfield in * *[Bou94 ]. Recall that given spaces A and X, X is called A-null if the natural inclusion X* * ,! map (A, X) is a weak equivalence. In this way one defines a functor PA : Spaces* * ! Spaces , coaugmented and idempotent, such that PAX is A-null for every X, and such that for every A-null space Y the coaugmentation induces a weak equivalen* *ce map (PAX, Y ) ' map (X, Y ). This functor can also be defined in the pointed ca* *tegory, and its main properties can be found in [Far95] and [Cha96 ]. In our case the role of A and X in the Chach'olski result will be played by B* *Z=p and BG, respectively. If C is the corresponding cofiber, from now on we shall d* *enote the BZ=p-nullification of C by P . As a consequence of previous results, describing CW BZ=pBG is equivalent to * *de- scribing P , which is in general a more accessible problem. In our particular s* *ituation it is convenient to assume G generated by order p elements because this implies* * au- tomatically (using Whitehead's Theorem) that P is a simply-connected space. Thus we can use fracture lemmas [BK72 , VI.8.1]; and as P is rationally trivial, we * *see that Q P ' Pp^, the product of the q-completions for all primes q. On the other han* *d, note that the additional hypothesis on the generation of G causes no restrictio* *n, as for every finite G there is a homotopy equivalence CW BZ=pB 1(G) ' CW BZ=pBG induced by the natural inclusion ([Flo07, Proposition 4.1]). It is easily proved that the q-torsion part of P (for every prime q 6= p) has* * the Q homotopy type of q6=pBG^q. Hence the difficult part is to compute Pp^. Accord* *ing to [FS07 , 3.2], Pp^coincides with the base of the Chach'olski fibration when t* *he total space is BG^p, so we are actually studying the BZ=p-cellularization of BG^p. M* *ore CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 41 precisely, if G is generated by order p elements, the equivalence CW BZ=p(BG^p* *) ' (CW BZ=pBG)^pis proved in [FS07 , Proposition 3.2]. This is a very peculiar pr* *operty for spaces which, in general, cannot be decomposed via an arithmetic square. In the philosophy of [BLO03 ], the homotopy theory of BG^pis codified in the* * p- fusion data of G. From this point of view it can be observed that the structure* * of Pp^ strongly depends on the minimal strongly closed p-subgroup A1(S) of S that cont* *ains the p-socle of S (called Cl S in [FS07 ]). In particular, it is a consequence o* *f the Puppe sequence and the definition of nullification that if A1(S) = S then Pp^is trivi* *al. This shows one should consider the case in which A1(S) is strictly contained in S. In [FS07 ] the latter case is studied under the additional assumption that NG* * (S) controls (strong) G-fusion in S. Recall that the normalizer controls (strong) f* *usion if for every subgroup P S and g 2 G such that gP g-1 S, there exist h 2 NG (S) and c 2 CG (P ), such that g = hc. Since A1(S) is normal in NG (S), [FS07 ] sh* *ows that Pp^is homotopy equivalent to the p-completion of B(NG (S)=A1(S)); this also shows, roughly speaking, that the structure of the mapping space map *(BZ=p, BG* *^p) depends heavily on A1(S). This result is used, in particular, to compute the BZ* *=2- cellularization of classifying spaces of simple groups (relying on Theorem 2.1 * *there). In the next subsection we describe the remaining cases, giving very explicit * *descrip- tions of Pp^in all the situations in which this space is not trivial. 4.2. The description of CWBZ=pBG. In [FS07 ] it was already anticipated that a complete description of the BZ=p-cellularization of classifying spaces of finit* *e groups would depend on a structure theorem for groups that contain a non-trivial stron* *gly closed p-subgroup that is not a Sylow p-subgroup; in that time such a classific* *ation was only known for p = 2 (Theorem 2.1). But even in this case there were exampl* *es of groups such that A1(S) 6= S and NG (S) does not control fusion in S _ in oth* *er words, groups that were beyond the scope of [FS07 ]. The key step missing in that paper was the role of the subgroup OA(G), whose importance was already evident _ in an independent group-theoretic context _ in [Foo97 ]. First, OA(G) is by definition a subgroup of A, so in the case A = A1(* *S) it is likely that it controls a "part" of the structure of CW BZ=pBG. Moreover, O* *A(G) is normal in G, so one might expect a strong relationship between the cofiber of Chach'olski fibration for BG^pand for B(G=OA(G))^p. The significance of OA(G) * *is also suggested by the fact that passing to the quotient G=OA(G) gives a conside* *rable 42 RAM'ON J. FLORES AND RICHARD M. FOOTE simplification in the p-fusion structure of this quotient group in the sense th* *at _ as we shall see in Section 5 _ there are normalizers (of subgroups/elements) that * *do not control fusion in G with images that do so in the quotient. These ideas, combi* *ned with the explication of the role of OA(G) in the classification result Theorem * *2.2, allow us to prove the main theorem, Theorem 4.2, which covers all extant cases * *for CW BZ=pBG, subsuming all by a uniform treatment. For the remainder of this subsection A = A1(S), and overbars will denote pass* *age __ __ __ to the quotient G ! G=OA(G). The normalizer of A in G will be denoted by N , and __ __ N =A will be called . We begin with a technical lemma that we will need in the proof of the main th* *eorem. Lemma 4.1. The group is p-perfect, i.e., has no normal subgroup of index p. Proof.By contradiction, assume that there is a nontrivial homomorphism ' : ! __ Z=p. Precomposing with the map BN ! B induced by the canonical projection __ __ yields an essential map BN ! BZ=p, which is trivial when restricted to BA . Mo* *re- __ __ __^ over, as N controls p-fusion in G , we have another nontrivial map BG^p! BN p ! BZ=p that is induced by a (also nontrivial) homomorphism _ : G ! Z=p. As the __ __ image of A under the natural projection G ! G is A, the definition of ff implie* *s that the composition BA ! BG ! BG^p! BZ=p is homotopically trivial. In particular, every element of order p of G should go to zero under the map _. But G is gener* *ated by order p elements, so _ should be nontrivial. This is a contradiction, and we* * are done. Alternatively, Theorem B of [Gol75 ] says that for any strongly closed A we h* *ave (G0\ S)A = (NG (A)0\ S)A. Thus if G has no normal subgroup of index p, neither does NG (A)=A. Corollary 2.5 therefore also shows is p-perfect since each Li* * is simple. Now we are in a position to prove the principal result of this section. Theorem 4.2. In the previous notation, Pp^is homotopy equivalent to the p-compl* *etion __ __ of the classifying space of N =A . W Proof.Let D be the cofiber of the Chach'olski map BZ=p ! BG^p(extended to all the homotopy classes of maps BZ=p !v BG^p), whose BZ=p-nullification is Pp* *^. We denote by h : BG^p! D the natural map, and by j : D ! Pp^the canonical CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 43 coaugmentation. Moreover, if A1 < A2, we will call iA1,A2the group inclusion A1* * ,! A2. We claim there are maps B ^p! Pp^and Pp^! B ^pthat are homotopy inverses to one another. g ^ __ __ __ First we define Pp^! B p. Recall that, as N controls G -fusion in S, the inc* *lusion __ __ __^ (Bi__N,__G)^p_^ N ,! G induces a homotopy equivalence BN p ' BG p (see for example [MP98 , Proposition 2.1]) . Now consider the diagram W v ^ h j ^ (4.1) BZ=p _____//BGp ____//_D____//Pp ~ Bss^p|| ~ __fflffl|~~ BG ^p ~ OO ~ ____^| ~ (BiN,G)p'| ~g0 g __| ~ BN ^ ~ p ~ ^ |~ Bpp |~ ~fflffl| B ^p and call ff the composition of all vertical maps. __ ff According to the definition of G and A, the composition BZ=p ! BG^p! B ^pis inessential for every map BZ=p ! BG^p. This implies that the composition ff O v* * is so, and hence there exists the lifting g0. As B ^pis BZ=p-null, g0also lifts t* *o g, and that is the map we were looking for. (BiA,G)^p jOh To construct f : B ^p! Pp^consider now the composition BA - ! BG^p-! Pp^. As the induced homomorphism of fundamental groups is trivial when restrict* *ed to every generator of A (by construction), the composition must be null-homotop* *ic on BA. In particular, it is inessential when precomposing with the map B(OA(G) \ BiOA(G)\A,A A) -! BA. As Pp^is p-complete (by [FS07 , 3.2]), and OA(G)\A is p-Sylow * *in OA(G) by definition of OA(G), we can apply [Dwy96 , Theorem 1.4] to obtain that* * the (BiOA(G),G)^p composition BOA(G) -! BG^p-! Pp^is again homotopically trivial. Then, 44 RAM'ON J. FLORES AND RICHARD M. FOOTE by Zabrodsky's Lemma [Dwy96 , 3.4], there exists a lifting f0 BOA(G) BiOA(G),G|| fflffl|jOhO(-)^p BG _______//Pp^:: vv Bss|| vv _fflffl|_f0v BG __ where (-)^pdenotes the p-completion BG ! BG^p. Now we have another map BN ! Pp^given by the composition __ __^ (Bi__N,__G)^p^(f0)^p^ BN -! BN p - ! BGp -! Pp , where we have used the fact that Pp^is p-complete. The next diagram is clearly commutative by construction: BiA,G BA ______________//BGD DD Bss|A|| Bss|| DDDD _fflffl|_Bi_A,__N_Bfflffl|!!Di__N,__G_ ^ BA ____//_BN_____//BG ____//_Pp . BiA,G jOhO(-)^p^ Note that the composition BA - ! BG - ! Pp is homotopically trivial, and __ Bi_A,__N_ f0 __ hence the composition BA -! BA -! BG -! Pp^is so. As A is by definition a quotient of A, every generator of the former comes from a generator from the la* *tter. __Bi_A,__N_ f0 This implies that the composition BA -! BG -! Pp^is also homotopically triv* *ial, and thus again by [Dwy96 , 3.4], the map (f0)^plifts to a map f: __^ (f0)^p ^ BN p ____//_Pp -== | --- | -- fflffl|f-- B ^p. This is the map f that we wanted; and we have another commutative diagram: __^ jOh ^ (4.2) BG p ____//_Pp -==- ff|| ---- fflffl|f-- B ^p. CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 45 It remains to prove that f O g ' IdP^pand g O f ' IdB ^p. In the first case, * *the universal property of the nullification functor implies that it is enough to pr* *ove that f O g0 is homotopic to the coaugmentation j. As Pp^is simply connected, we only need to prove that f O g0' j in the unpointed category. Hence we can use the lo* *ng exact sequence of the cofibration BG^p!h D ! _ BZ=p to establish that f O g0' j if and only if f O g0O h ' j O h. According to diagram 4.1, g0O h is homotopic * *to ff, and by diagram 4.2, f O ff ' j O h, so we are done. To see that g O f is homotopic to the identity of B ^p, a repeated applicatio* *n of the universal property of the quotient shows that it is enough to prove that gOf Of* *f ' ff. By Lemma 4.1 and [BK72 , II.5], B ^pis simply connected, and we can apply the prev* *ious arguments to ensure that it is enough to find the homotopy in the unpointed cat* *egory. Again by diagram 4.2 the latter is homotopic to g O j O h, and has the same hom* *otopy class as ff by diagram 4.1. So the statement is proved. When we combine the previous statement with [Flo07, Proposition 4.14] and [FS* *07 , Theorem 2.5] we obtain a complete description of CW BZ=pBG for every p and eve* *ry finite group G. Theorem 4.3. Let G be a finite group generated by its elements of order p, let * *S 2 Sylp(G), and let A = A1(S) be the minimal strongly closed subgroup of S contain* *ing 1(S). Then the BZ=p-cellularization of BG has one the following shapes: (1) If G = S is a p-group then BG is BZ=p-cellular. (2) If G is not a p-group and A = S then CW BZ=pBG is the homotopy fiber of Q the natural map BG ! q6=pBG^q. (3) If G is not a p-group and A 6= S then CW BZ=pBG is the homotopy fiber of __ __ Q the map BG ! B(N__G(A )= A)^px q6=pBG^q. __ __ Theorems 2.1 and 2.2 and Corollary 2.5 determine NG (A )=A , whose structure * *is very rigid and depends on a restricted set of well-known simple groups. It is * *very likely that an analogous classification can be obtained exactly in the same way* * for CW BZ=prBG, r > 1, but we have restricted ourselves to the case r = 1 for the * *sake of simplicity (cf. also [CCS , Theorem 3.6]). In the cases where OA(G) = 1 and A E G _ which are implicit in the computatio* *ns _ the BZ=p-cellularization of BG^pis the homotopy fiber of the natural map BG^p! B(G=A)^p. It is then tempting to identify CW BZ=pBG with BA. But this would me* *an, 46 RAM'ON J. FLORES AND RICHARD M. FOOTE in particular, that map *(BZ=p, BG^p) would be discrete. However, an analysis o* *f the fibration map *(BZ=p, BG^p) ! map (BZ=p, BG^p) ! BG^p together with the description of its total space _ which is given, for example,* * in [BK02 , Appendix] _ shows that map *(BZ=p, BG^p) is non-discrete in general, and then usually CW BZ=pBG is not an aspherical space. It is conceivable that our results can also have interesting consequences fro* *m the point of view of homotopical representations of groups. In [FS07 , Section 6] t* *he results on cellularization gave rise to specific examples of nontrivial maps BG ! BU(n)* *^p that enjoyed two particular properties: they did not come from group homomorphi* *sms G ! U(n), and they were trivial when precomposing with any map BZ=p ! BG. While there are a number of examples in the literature with the first feature (* *see for example [BW95 ] or [MT89 ]), no representations were known at this point for * *which the second property holds. The classification results of this paper give hope o* *f finding a systematic and complete treatment of all these kinds of representations. We p* *lan to undertake this task in a separate paper. In the next section we show the applicability of our results by computing the BZ=p-cellularization of various specific families of classifying spaces. We hav* *e chosen the simple groups (as they have shown their cornerstone role in the computation* * of CW BZ=pBG), certain split extensions that signaled there was something beyond * *the results of [FS07 ], and certain nonsplit extensions of G2(q) that illuminate th* *e roles of the normalizers of A and S in the BZ=p-cellular context. 5.Examples Throughout this section p is any prime, G is a finite group possessing a nont* *rivial Sylow p-subgroup S and A1(S), as before, denotes the unique smallest strongly c* *losed (with respect to G) subgroup of S that contains 1(S). In Theorem 4.3 a descrip* *tion of the BZ=p-cellularization of BG for every group G and every prime p is given.* * In this section we describe some families of concrete examples for which CW BZ=pB* *G is interesting. We begin with the case of simple groups. 5.1. Simple groups. In [FS07 , Corollary 5.7] the BZ=2-cellularization of the c* *las- sifying spaces of all the simple groups was computed. In this section we use t* *he classification, Theorem 2.2, to show that for every prime p and every simple gr* *oup CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 47 G, CW BZ=pBG is included in cases (ii) and (iii) of Theorem 4.3. The key resul* *t is the following immediate consequence of Theorems 2.1 and 2.2 (where OA(G) = 1 by the simplicity of G): Corollary 5.1. Let G be a simple group in which A1(S) 6= S. Then G is isomorphic to one of the groups Li that appear in the conclusions of Theorems 2.1 and 2.2. Next, recall that if G is the simple group G2(q) for some q with (q, 3) = 1, * *then we showed in the proof of Proposition 2.12 (and at the end of the proof of Lemma 3* *.4) that S = 1(S). Then by Corollary 2.4, in all cases in Corollary 5.1 the normal* *izer of S controls strong fusion in S. Thus Theorem 4.3 yields the following characteri* *zation: Proposition 5.2. Let G be a simple group, let p a prime and let S be a Sylow p-subgroup of G. Then CW BZ=pBG has one of the following two structures: (1) If A1(S) = S, then CW BZ=pBG is the homotopy fiber of the natural map Q BG ! q6=pBG^q. (2) If A1(S) 6= S, then we have a fibration Y CW BZ=pBG ! BG ! B(NG (S)=A1(S))^px BG^q. q6=p Note that the inclusion NG (S) ,! NG (A1(S)) induces a homotopy equivalence BNG (S)^p ' BNG (A1(S))^pwhen NG (S) (and then NG (A1(S))) controls p-fusion in S. This happens for every simple group in the second case of the previous proposition, and in particular a comparison of Chach'olski fibrations for BNG (* *S)^p and BNG (A1(S))^p(which are homotopy equivalent) gives that the induced map B(NG (S)=A1(S))^p' B(NG (A1(S))=A1(S))^pis also a homotopy equivalence. In the following we use Theorem 4.3 to describe explicitly the BZ=p-cellulari* *zation of the classifying spaces of the groups of the second kind in the previous stat* *ement, which turn out to be some of the ones that appear in the classification. With * *the exception of the groups of Lie type in characteristic 6= p, the Sylow-p normali* *zers of the simple groups appearing in the conclusions to Theorems 2.1 and 2.2 are desc* *ribed explicitly in Proposition 2.12. We therefore add here only some observations on* * the structure of the normalizers in the remaining case. Let G be a group of Lie type over a field of characteristic r 6= p and suppose the Sylow p-subgroup S of G is abelian but not elementary abelian (here p is od* *d). The overall structure of NG (S) is governed by the theory of algebraic groups, * *as 48 RAM'ON J. FLORES AND RICHARD M. FOOTE invoked in the proof of Proposition 2.11. Recapping from that argument: since t* *he Schur multiplier of G is prime to p we may work in the universal version of G to __ describe NG (S). Let G be the simply connected universal simple algebraic gro* *up over the algebraic closure of Fr, and let oe be a Steinberg endomorphism whose * *fixed __ points equal G. In the notation of [SS70 ], p is not a torsion prime for G , so* * by 5.8 therein C__G(S) is a connected, reductive group whose semisimple component is s* *imply connected. The general theory of connected, reductive algebraic groups gives t* *hat ____ __ __ C__G(S) = Z L, where Z is the connected component of the center of C__G(S), L i* *s the __ __ semisimple component (possibly trivial), and Z \ L is a finite group. Furthermo* *re, __ L is a product of groups of Lie type over the algebraic closure of Fr of smalle* *r rank __ * * __ than G . It follows that C__G(S) is a commuting product of the fixed points of * *oe on Z __ and L, i.e., C__G(S) = C__Z(oe)C__L(oe) where S C__Z(oe) is an abelian group (a finite torus) and C__L(oe) is either * *solvable or a product of finite Lie type groups in characteristic r. To complete the generic description of NG (S) we invoke additional facts from* * [SS70 ] * * __ and [GLSv3 , Section 4.10]. As above, S is contained in a oe-stable maximal tor* *us T1, __ __ where T 1is obtained from a oe-stable split maximal torus T by twisting by some __ __ __ element w of the Weyl group W = N__G(T )=T of G . Since S is characteristic i* *n the ____ finite torus T1 = (T1)oeit follows that NG (S)=CG (S) ~=NG (T1)=T1. In most cas* *es, by 1.8 of [SS70 ] or Proposition 3.36 of [Ca85 ] we have NG (T1)=T1 ~=Woe~=CW (w) * *(see also [GLSv3 , Theorem 2.1.2(d)] and the techniques in the proof of Theorem 4.10* *.2 in that volume). In the special case where G is a classical group (linear, unitary, symplectic* *, orthog- onal) the normalizer of S can be computed explicitly by its action on the under* *lying natural module, V , as described in the proof of Lemma 3.4. In the notation of * *this lemma, the semisimple component of order prime to p comes from the normal sub- group Isom(V0) in Isom(V ), where V0 = CV (S), and S is the direct product of t* *he cyclic groups S \ Isom(Vi) for i = 1, 2, . .,.s. The Weyl group normalizing S a* *cts as the symmetric group Ss permuting the subgroups Isom(Vi). The orders of the cent* *ral- izer and normalizer of a (cyclic) Sylow p-subgroup in each subgroup Isom(Vi) de* *pend on p and the nature of G _ Chapter 3 of [Ca85 ] gives techniques for computing * *these. For an easy explicit example of this let G = SLn+1(q) where q = rm and p > n * *+ 1, fi and assume p fiq -1. In this case we may choose S contained in the group of dia* *gonal CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 49 matrices T of determinant 1, which is an abelian group of type (q-1, . .,.q-1) * *of rank n (here T is the split torus). In this case T = CG (S) and NG (S) = NG (T ) = T* * W , where W ~=Sn+1 is the group of permutation matrices permuting the entries of matrices in T in the natural fashion (as the "trace zero" submodule of the natu* *ral action on the direct product of n + 1 copies of the cyclic group of order q - 1* *). To obtain the Sylow p-normalizer in the simple group P SLn+1(q) factor out the sub* *group of scalar matrices of order (n + 1, q - 1). 5.2. Split extensions. In this subsection we consider some non-simple groups. H* *ere we give some explicit examples of BZ=p-cellularization of split extensions whic* *h are beyond the scope of [FS07 ], and show the usefulness of Theorems 4.2 and 4.3. In [FS07 ] the BZ=p-cellularization of BG is described when G is generated by elements of order p, A1(S) is a proper subgroup of S, and the normalizer of S c* *ontrols strong fusion in S. No example was given there of a group for which the first * *two conditions hold but not the third. George Glauberman suggested an example of a group of the latter type: a wreath product (Z=2) o Sz(2n). In this section * *we generalize this example, showing that many split extensions for which these con* *ditions hold can be constructed. The computation of the cellularization of their classi* *fying space is then easy from Corollaries 2.4 and 2.5. This construction demonstrates* * that __ __ __ __ even when N__G(S ) (or N__G(A )) controls G -fusion in S, where overbars denote* * passage to G=OA(G), it need not be the case that NG (A) controls fusion in S (or in A), __ _______ even when N__G(A ) = NG (A). This highlights the importance of "recognizing" t* *he subgroup OA(G) as well as the isomorphism types of the components of G=OA(G) in our classifications. Proposition 5.3. Let R be any group that is not a p-group but is generated by elements of order p. Assume also that A1(T ) 6= T for some Sylow p-subgroup T * *of R. Let E be any elementary abelian p-group on which R acts in such a way that R=CR(E) is not a p-group. Let G be the semidirect product E o R, and let S = ET be a Sylow p-subgroup of G. Then G is generated by elements of order p, A1(S) 6* *= S, and NG (S) does not control fusion in S. Proof.Note that the split extension G = ER is clearly generated by elements of order p since both E and R are. Also, A1(S) contains E, and by Lemma 2.8, since the extension is split we obtain A1(S)=E ~=A1(T ) < T , so A1(S) 6= S. It remai* *ns to show that NG (S) does not control fusion in S. 50 RAM'ON J. FLORES AND RICHARD M. FOOTE Let 0 = E0 < E1 < . .<.En-1 < En = E be a chief series through E, so that each factor Ei=Ei-1is an irreducible FpR-module. If each such factor is one-dimensio* *nal, then R is represented by upper triangular matrices in its action on E. Since R is generated by elements of order p, it must be represented by unipotent matric* *es, hence R=CR(E) is a p-group, a contradiction. Thus there is some chief factor Ei=Ei-1 that is not one-dimensional. If a Sy* *low normalizer controlled fusion in S, then by Lemma 2.8 the same would be true in the quotient group G=Ei-1; we show this is not the case. To do so, we may pass * *to the quotient and therefore assume E1 is a minimal normal, noncentral subgroup of G. Now Z1 = Z(S) \ E1 6= 1 and Z1 is invariant under NG (S). However, R acts irreducibly and nontrivially on E1 and R is generated by conjugates of S, so Z1* * 6= E1 and hence Z1 is not R-invariant. Thus for some z 2 Z1 and g 2 G we must have zg 2 E1 - Z1, which shows NG (S) does not control fusion in S. This proposition can be invoked to create a host of examples: Let R be any of the simple groups Li (or their quasisimple universal covers) in the conclusion * *to Theorem 2.2 and let E be an FpR-module on which R acts nontrivially (for exampl* *e, any nontrivial permutation module). More specifically, for p odd let q be any p* *rime fi power such that p2 fiq - 1, so that Sylow p-subgroups of R = SL2(q) are cyclic of order p2 (for example, p = 3 and q = 19). Then R permutes the q + 1 lines in a 2-dimensional space over Fq, and so permutes q + 1 basis vectors in a q + * *1- dimensional vector space E over Fp. Then G = E o R gives a specific realization* * for Proposition 5.3. If G = E oR is any group satisfying the conditions of the previous propositio* *n with R simple and A = EA1(T ), in the notation of Section 2 it is clear that E = OA(* *G). As the extension is split, the canonical projection G ! R sends A1(S) to A1(T ). Then, according to Theorem 4.2, the BZ=p-cellularization of BG^phas the homotopy type of the fiber of the composition BG^p-! BR^p-! B(NR(A1(T ))=A1(T ))^p. As R is simple, the base of the fibration was studied in the previous subsectio* *n. Building on the preceding example where R = SL(2, q) for any prime power q fi such that p2 fiq - 1: then T may be represented by diagonal matrices over Fq, so is cyclic of order pn = |q - 1|p; moreover, CR(T ) is the group of all diago* *nal matrices of determinant 1, hence is cyclic of order q - 1. In particular, A1(T* * ) = CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 51 1(T ) ~= Z=p. Furthermore, NR(T ) = NR(A1(T )) is of index 2 in CR(T ) and an involution in NR(T ) inverts CR(T ). Thus NR(A1(T ))=A1(T ) is isomorphic to t* *he dihedral group of order 2(q - 1)=p. Again comparing Chach'olski fibrations, we * *obtain that B(NR(A1(T )))=A1(T ))^pis homotopy equivalent to B(NR(T )=A1(T ))^p. 5.3. Exotic extensions of G2(q)G2(q)G2(q). When G is the simple group G2(q) for* * some q with (q, 3) = 1, although a Sylow 3-subgroup S contains a strongly closed subgr* *oup A = Z(S) of order p = 3, when we impose the additional hypothesis that our stro* *ngly closed subgroup must contain all elements of order 3 the strongly closed subgro* *up A does not arise in our considerations because S = 1(S). For the same reason, if G = ER is any split extension of R = G2(q) by an elementary abelian 3-group and S = ET for T 2 Syl3(R), then again S = 1(S) = A1(S). In this subsection we describe a family of extensions that we call "half-split" in the sense that the* *y split over a certain conjugacy class of elements of R but do not split over another. * *In this way we construct extensions G of R = G2(q) by certain elementary abelian 3-grou* *ps E such that for S 2 Syl3(G) we have 1(S)=E mapping onto the strongly closed subgroup of order 3 in a Sylow 3-subgroup S=E of G2(q). In particular, these "e* *xotic" extensions show that the exceptional case of Corollary 2.4 cannot be removed: w* *hen fi 9 fiq2 - 1 these groups G are generated by elements of order 3, have A1(S) 6= S* *, but NG=E(S=E) does not control fusion in S=E (here E = OA(G) where A = A1(S)). The following general proposition will construct such extensions. Proposition 5.4. Let p be a prime dividing the order of the finite group R and * *let X be a subgroup of order p in R. Then there is an FpR-module E and an extension 1 -! E -! G -! R -! 1 of R by E such that the extension of X by E does not split, but the extension of Z by E splits for every subgroup Z of order p in R that is not conjugate to X. * * In particular, for nonidentity elements x 2 X and z 2 Z every element in the coset* * xE has order p2 whereas zE contains elements of order p in G. Proof.Let E0 be the one-dimensional trivial FpX-module. By the familiar cohomol- ogy of cyclic groups ([Bro82 ], Section III.1): (5.1) H2(X, E0) ~=Z=pZ 52 RAM'ON J. FLORES AND RICHARD M. FOOTE and a non-split extension of X by E0 is just a cyclic group of order p2. Now let E = CoindRXE0 = Hom ZX (ZR, E0) be the coinduced module from X to R (which is isomorphic to the induced module E0 FpX FpR in the case of finite groups), so that E has Fp-dimension 1_p|R|. * *By Shapiro's Lemma ([Bro82 ], Proposition III.6.2) (5.2) H2(R, E) ~=H2(X, E0). Thus by (5.1) there is a non-split extension of R by E _ call this extension gr* *oup G and identify E as a normal subgroup of G with quotient group G=E = R. The isomorphism in Shapiro's Lemma, (5.2), is given by the compatible homomor- phisms ' : X ,! R and ss : CoindRXE0 ! E0, where ss is the natural map ss(f) = * *f(1). In particular, this isomorphism is a composition * 2 H2(R, E) -res!H2(X, E) -ss!H (X, E0). Thus the 2-cocycle defining the non-split extension group G, which maps to a no* *n- trivial element in H2(X, E0), by restriction gives a non-split extension of X b* *y E as well. For any subgroup Z of R of order p with Z not conjugate to X, by the Mackey decomposition for induced representations M -1 (5.3) ResRZIndRXE0 = IndZZ\gXg-1ResgXgZ\gXg-1gE0 g2R where R is a set of representatives for the (Z, X)-double cosets in R. By hypot* *hesis, Z \ gXg-1 = 1 for every g 2 R, hence each term in the direct sum on the right h* *and side is an FpZ-module obtained by inducing a one-dimensional trivial Fp-module * *for the identity subgroup to a p-dimensional FpZ-module, i.e., is a free FpZ-module* * of rank 1. (Alternatively, E is the Fp-permutation module for the action of R by * *left multiplication on the left cosets of X; by the fusion hypothesis, Z acts on a b* *asis of E as a product of disjoint p-cycles with no 1-cycles.) This shows E is a free FpZ* *-module, and hence the extension of Z by E splits. This completes the proof. The pth-power map on elements in the lift of X to G can be described more pre* *cisely. By the Mackey decomposition in (5.3) inducing from X but rather restricting to X CELLULARIZATION VIA STRONGLY CLOSED SUBGROUPS 53 instead of Z, or by direct inspection of the action of X on the Fp-permutation * *module E, we see that E decomposes as an FpX-module direct sum as E = E1 E2, where E1 is a trivial FpX-module and E2 is a free FpX-module. Since X splits ov* *er the free summand E2, we see that X does not split over E1, and hence XE1 ~=(Z=p2) x Z=p x . .x.Z=p with E1 = 1(XE1). Thus for every element x in G - E mapping to an element of X in G=E, xp has a nontrivial component in E1. One may also observe that by taking direct sums we can arrange more generally* * that if X1, X2, . .,.Xn are representatives of the distinct conjugacy classes of sub* *groups of order p in R, then for any i 2 {1, 2, . .,.n} there is an FpR-module E and an e* *xtension of R by E such that in the extension group each of X1, . .,.Xisplits over E but* * none of Xi+1, . .,.Xn do. We are particularly interested in the case R = G2(q) with p = 3 and (q, 3) = * *1. The normalizer of a Sylow 3-subgroup of R is described in Proposition 2.12: Let T 2 Syl3(R) and let Z = Z(T ) = < z >. In the notation preceding Proposition 2.* *10, fi fi NR(Z) ~=SLffl3(q) . 2 according as 3 fiq - ffl. Moreover, if 9 fiq - ffl then N* *R(T ) does not control fusion in T : all elements of order 3 in T - Z are conjugate in CR(Z) w* *hereas by Proposition 2.12, NR(T )=T has order 4 for this congruence of q. Thus BNR(T * *)^3 is not homotopy equivalent to BG2(q)^3. Now consider the extension group G constructed in Proposition 5.4 with p = 3, R = G2(q), Z = < z > and X = < x > for any x 2 T -Z of order 3. Let S 2 Syl3(G)* * with S mapping onto T in G=E ~=R. Since Proposition 2.12 shows all elements of order 3 in T - Z are conjugate to x but not to z, the structure of the extension impl* *ies that A = 1(S) = A1(S) contains E and maps to Z in S=E. Thus OA(G) = E __ __ and A = Z . By Corollary 2.4, the normalizer of Z in R = G2(q) controls 3-fusi* *on in G2(q), so in particular SL*3(q) has the same mod 3 cohomology as G2(q), where SL*3(q) denotes the group SLffl3(q) together with the outer (graph) automorphis* *m of order 2 inverting its center (NR(Z) ~=SL*3(q)). On the other hand, Z is normal* * in SL*3(q), and SL*3(q)=Z is isomorphic to P SL*3(q). Hence, by Theorem 4.2, the B* *Z=3- cellularization of (BG)^3is the homotopy fibre of the map (BG)^3! BP SL*3(q)^3,* * but fi not of the map (BG)^3! B(NR(T )=T )^3when 9 fiq2 - 1. 54 RAM'ON J. FLORES AND RICHARD M. FOOTE From this computation one can deduce that the object that determines the BZ=* *p- nullification of the cofibre of the Chach'olski map in the case of finite group* *s is the __ normalizer of A, and not the Sylow normalizer, as might be inferred from the pa* *rtic- ular cases studied in [FS07 ]. This example also highlights the importance of h* *aving a classification of all groups possessing a nontrivial strongly closed p-subgroup* * that is not Sylow _ not just the simple groups having such a subgroup that contains 1(* *S) _ since the subgroup A1(S) does not pass in a transparent fashion to quotients. In conclusion, some interesting open questions remain. We have characterized * *with precision CW BZ=pBG for every finite G, and in the course of the proof we have* * also described CW BZ=pBG^pwhen G is generated by order p elements. 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Steinberg, Conjugacy classes. in Seminar on Algebraic * *Groups and Related Finite Groups ed. by A. Borel et. al., Springer Lecture Notes #13* *1, Springer-Verlag, Berlin, 1970. [Suz62]M. Suzuki, On a class of doubly transitive groups. Ann. of Math., 75(196* *2), 105-145. [Wa66]H. N. Ward, On Ree's series of simple groups. Trans. Amer. Math. Soc., 12* *1(1966), 62-89. [Wi98]R. A. Wilson, Some subgroups of the Thompson group. J. Australian Math. S* *oc., 44(1998), 17-32. Ram'on J. Flores Departamento de Estad'istica, Universidad Carlos III de Madrid, C/ Madrid 126 E - 28903 Colmenarejo (Madrid) _ Spain e-mail: rflores@est-econ.uc3m.es Richard M. Foote Department of Mathematics and Statistics, University of Vermont, 16 Colchester Avenue, Burlington, Vermont 05405 _ U.S.A. e-mail: foote@math.uvm.edu