Strongly closed subgroups of finite groups
aa r X i v : . [ m a t h . G R ] S e p STRONGLY CLOSED SUBGROUPS OF FINITE GROUPS
RAM ´ON J. FLORES AND RICHARD M. FOOTE
Abstract.
This paper gives a complete classification of the finite groups thatcontain a strongly closed p -subgroup for p any prime. Introduction
For any finite group G and subgroup S we say two elements of S are fused in G ifthey are conjugate in G but not necessarily in S . This concept has played a centralrole in group theory and representation theory, particularly in the case when S is aSylow p -subgroup of G for p a prime. A subgroup A of S is called strongly closed in S with respect to G if for every a ∈ A , every element of S that is fused in G to a lies in A ; in other words, a G ∩ S ⊆ A , where a G denotes the G -conjugacy classof a . It is easy to verify that if A is a p -subgroup, then A is strongly closed in aSylow p -subgroup if and only if it is strongly closed in N G ( A ), so the notion of strongclosure for a p -subgroup does not depend on the Sylow subgroup containing it. For a p -group A we therefore simply say A is strongly closed. Seminal works in the theoryof strongly closed 2-subgroups are the celebrated Glauberman Z ∗ -Theorem, [Gla66],and Goldschmidt’s theorem on strongly closed abelian 2-subgroups, [Gol74]. The Z ∗ -Theorem proved that if A is strongly closed and of order 2, then A ≤ Z ( G ), where theoverbars denote passage to G/O ′ ( G ). Goldschmidt extended this by showing that if A is a strongly closed abelian 2-subgroup, then h A G i is a central product of an abelian2-group and quasisimple groups that either have a BN -pair of rank 1 or have abelianSylow 2-subgroups. These two theorems, in particular, played fundamental roles inthe study of finite groups, especially in the Classification of the Finite Simple Groups.The purpose of this paper is to give a classification of all finite groups containing astrongly closed p -subgroup for an arbitrary prime p (not assuming the strongly closedsubgroup is abelian). Date : August 22, 2018.The first author was supported by MEC grant MTM2004-06686.
The concept of strong closure has important ramifications beyond finite grouptheory. In particular, it is intimately connected to Puig’s formulation of fusion systems (or Frobenius categories), which evolved from the modular representation theory offinite groups: To each p -block of a finite group one can associate a (saturated) fusionsystem. Puig’s axiomatic approach provided the formalism necessary to study fusionin a context which subsumes, as a special case, the natural fusion system arisingfrom pairs ( G, S ), where G is a finite group and S is a Sylow p -subgroup of G .The concept of strong closure extends in an obvious way to abstract fusion systemsand plays a critical role therein: If F is a fusion system on a p -group S , then thehomomorphic images of F are in bijective correspondence with the strongly closedsubgroups of S . Fusion systems were further refined by Broto, Levi, and Oliver in[BLO03] to create the class of p -local finite groups (see also [Asc07], [BCGLO07],[BLO07] and [Lin06]). Oliver then used this approach to prove that the homotopytype of the p -completed classifying space of a finite group G is uniquely determinedby the saturated fusion system ( G, S ), where S is a Sylow p -subgroup of G . Thusstrong closure and its extensions to fusion systems and p -local finite group theoryalso has significant ramifications in deep and currently very active areas of modularrepresentation theory and algebraic topology.This paper is also the group-theoretic result needed for a classification theorem inhomotopy theory, which was the original impetus for our joint work. Groups contain-ing a strongly closed 2-subgroup were characterized earlier in [Foo97], and that the-orem formed the underpinning of a complete description of the B Z / B Z /p -cellularization of classifyingspaces for odd primes p , we needed the classification of finite groups containing astrongly closed p -subgroup for odd p — this is the main theorem herein. The com-plete description of the cellular structure (with respect to B Z /p ) of classifying spacesfor all finite groups and all primes p is then established in the separate paper [FlFo08].Our two classifications, the latter relying on the former, epitomize the rich interplaybetween their subject areas that has historically been evident and is currently evenmore vibrant.A curious application of strong closure to ordinary representation theory and num-ber theory appears in [Foo97a]. TRONGLY CLOSED SUBGROUPS 3
Finally, although the techniques used in this paper are purely group-theoretic, theunderlying fusion arguments provide deeper insight into topological considerations inour second classification. Indeed, the marriage of these elements is seen in high reliefin Section 4 where we explore more explicit configurations that give rise in [FlFo08]to interesting — what might be called exotic — classifying spaces.
Acknowledgements . We thank Carles Broto, David Dummit, Bob Oliver andJ´erˆome Scherer for helpful discussions, and also George Glauberman for providinga motivating example. We thank Michael Aschbacher for sharing his lecture notesand broad perspective, and we acknowledge that a number of the results in Section 3were also proven independently by him.1.1.
Statement of Results.
To describe the main results we introduce some new notation. Henceforth p is anyprime, S is a Sylow p -subgroup of the finite group G and A is a subgroup of S . Ingeneral let R be any p -subgroup of G . If N and N are normal subgroups of G with R ∩ N i ∈ Syl p ( N i ) for both i = 1 ,
2, then R ∩ N N is a Sylow p -subgroup of N N .Thus there is a unique largest normal subgroup N of G for which R ∩ N ∈ Syl p ( N );denote this subgroup by O R ( G ). Thus R is a Sylow p -subgroup of h R G i if and only if R ≤ O R ( G ) . Note that O p ′ ( G/ O R ( G )) = 1; in particular, if R = 1 is the identity subgroup then O ( G ) = O p ′ ( G ). In general, R O R ( G ) / O R ( G ) does not contain the Sylow p -subgroupof any nontrivial normal subgroup of G/ O R ( G ); in other words, O R ( G ) = 1, whereoverbars denote passage to G/ O R ( G ). Throughout the paper we freely use the ob-servation that strong closure passes to quotient groups (cf. Lemma 2.3), so whenanalyzing groups where R
6≤ O R ( G ) we may factor out O R ( G ). With this in mind,the classification for strongly closed 2-subgroups from [Foo97] is as follows: Theorem 1.1.
Let G be a finite group that possesses a strongly closed 2-subgroup A .Assume A is not a Sylow 2-subgroup of h A G i , and let G = G/ O A ( G ) . Then A = 1 and h A G i = L × L × · · · × L r , where each L i is isomorphic to U (2 n i ) or Sz (2 n i ) for some n i , and A ∩ L i is the center of a Sylow 2-subgroup of L i . The classification for p odd, which is the principal objective of the paper, yields amore diverse set of “obstructions” with added “decorations” as well. RAM ´ON J. FLORES AND RICHARD M. FOOTE
Theorem 1.2.
Let p be an odd prime and let G be a finite group that possesses astrongly closed p -subgroup A . Assume A is not a Sylow p -subgroup of h A G i , and let G = G/ O A ( G ) . Then A = 1 and (1.1) h A G i = ( L × L × · · · × L r )( D · A F ) where r ≥ , each L i is a simple group, and A i = A ∩ L i is a homocyclic abelian group.Furthermore, D = [ D, A F ] is a (possibly trivial) p ′ -group normalizing each L i , and A F is a (possibly trivial) abelian subgroup of A of rank at most r normalizing D and each L i and inducing outer automorphisms on each L i , and the extension ( A · · · A r ) : A F splits. Each L i belongs to one of the following families: (i) L i is a group of Lie type in characteristic = p whose Sylow p -subgroup isabelian but not elementary abelian; in this case the Sylow p -subgroup of L i ishomocyclic of the same rank as A i but larger exponent than A i ; here D/ ( D ∩ L i C G ( L i )) is a cyclic p ′ -subgroup of the outer diagonal automorphism groupof L i , and A F /C A F ( L i ) acts as a cyclic group of field automorphisms on L i . (ii) L i ∼ = U ( p n ) or Re (3 n ) is a group of BN -rank 1 ( p = 3 with n odd and ≥ inthe latter family); in the unitary case A i is the center of a Sylow p -subgroup of L i (elementary abelian of order p n ), and in the Ree group case A i is either thecenter or the commutator subgroup of a Sylow 3-subgroup (elementary abelianof order n or n respectively); in both families D and A F act trivially on L i . (iii) L i ∼ = G ( q ) with ( q,
3) = 1 ; here | A i | = 3 and both D and A F act trivially on L i . (iv) L i is one of the following sporadic groups, where in each case A i has primeorder, and both D and A F act trivially on L i : ( p = 3 ) : J , ( p = 5 ) : Co , Co , HS , M c , ( p = 11 ): J . (v) L i ∼ = J , p = 3 , and A i is either the center or the commutator subgroup of aSylow 3-subgroup (elementary abelian of order 9 or 27 respectively); here D and A F act trivially on L i . Remark.
After factoring out O A ( G ) — so that overbars may be omitted — the proofof Theorem 1.2 shows that F ∗ ( G ) = L × · · · × L r , and (1.1) may also be written as h A G i ∼ = (( L × · · · × L i ) D × L i +1 × · · · × L j ) A F × ( L j +1 × · · · × L r ) TRONGLY CLOSED SUBGROUPS 5 where L , . . . , L i are the components of type P SL or P SU over fields of characteristic = p , L i +1 , . . . , L j are other groups listed in conclusion (i) (but not linear or unitary),and L j +1 , . . . , L r are the components of types listed in (ii) to (v). Furthermore,assume G = h A G i and let A ≤ S ∈ Syl p ( G ) and S ∗ = S ∩ F ∗ ( G ). Then we maychoose D generically as [ O p ′ ( C G ( S ∗ )) , S ], which is a p ′ -group normalized by S andcentralized by the Sylow p -subgroup S ∗ of L · · · L r .An easy example where both D and A F are nontrivial is provided at the outset ofSection 4.Conversely, observe that any finite group that has a composition factor of one ofthe above types for L i possesses a strongly closed p -subgroup that is not a Sylow p -subgroup of its normal closure in G . More detailed information about the structureof the Sylow p -subgroups and their normalizers for the simple groups L i appearingin the conclusion to this theorem is given from Proposition 2.4 through Corollary 2.8following.Theorem 1.2 is derived at the end of Section 3 as a consequence of the next result,which is the minimal configuration whose proof appears in Section 3. Theorem 1.3.
Assume the hypotheses of Theorem 1.2. Assume also that A is aminimal strongly closed subgroup of G , i.e., no proper, nontrivial subgroup of A isalso strongly closed. Then the conclusion of Theorem 1.2 holds with the additionalresults that A is elementary abelian, D = 1 , A F = 1 , and G permutes L , . . . , L r transitively (hence they are all isomorphic). Some important consequences needed for our results on cellularization of classifyingspaces in [FlFo08] are the following.
Corollary 1.4.
Let p be any prime, let G be a finite group containing a strongly closed p -subgroup A , let S be a Sylow p -subgroup of G containing A , and let G = G/ O A ( G ) .Assume that G is generated by the conjugates of A . Then N G ( A ) controls strong G -fusion in S . Furthermore, if p = 3 or if G does not have a component of type G ( q ) with (cid:12)(cid:12) q − , then N G ( S ) controls strong G -fusion in S . In Section 4.3 we demonstrate that the exceptional case to the stronger conclusionin the last sentence of Corollary 1.4 is unavoidable, even if we impose the conditionthat Ω ( S ) ≤ A : we construct examples of groups G generated by conjugates of astrongly closed subgroup A containing Ω ( S ) and G/ O A ( G ) ∼ = G ( q ) where N G ( S )does not control fusion in S . RAM ´ON J. FLORES AND RICHARD M. FOOTE
The next result facilitates computation of N G ( A ) in groups satisfying the conclusionto the preceding corollary. Corollary 1.5.
Assume the hypotheses of preceding corollary and the notation ofTheorem 1.2. For each i let C i = C G ( A F ) ∩ N L i ( A i ) and S i = S ∩ L i . Then N G ( A ) / A = ( S C /A ) × ( S C /A ) × · · · × ( S r C r /A r ) . In particular, if L i is a component on which A F acts trivially — which is the case forall components in conclusions (ii) to (v) of Theorem 1.2 — the i th direct factor abovemay be replaced by just N L i ( A i ) /A i (and this applies to all factors if A F = 1 ). The proof of Theorem 1.3 relies on the Classification of Finite Simple Groups. Wereduce to the case where a minimal counterexample, G , is a simple group havinga 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 thefamilies of simple groups to determine precisely when this happens.We note that “most” simple groups do possess a strongly closed p -subgroup that isproper in a Sylow p -subgroup, that is, conclusion (i) of Theorem 1.2 is the “genericobstruction” in the following sense. Let L n ( q ) denote a simple group of Lie type and BN -rank n over the finite field F q with ( q, p ) = 1. As we shall see in Section 2, for allbut the finitely many primes dividing the order of the Weyl group of the untwistedversion of L n ( q ) the Sylow p -subgroups of L n ( q ) are homocyclic abelian. Furthermore,the order of L n ( q ) can be expressed as a power of q times factors of the form Φ m ( q ) r m for various m, r m ∈ N , where Φ m ( x ) is the m th cyclotomic polynomial. Then byProposition 2.4 below, if m is the multiplicative order of q (mod p ), then p divides Φ m ( q ) and the abelian Sylow p -subgroup of L n ( q ) is homocyclic of rank r m andexponent | Φ m ( q ) | p . In particular it is not elementary abelian whenever p (cid:12)(cid:12) Φ m ( q ).For example, this is the case in the groups P SL n +1 ( q ) whenever p > n + 1 and p divides q m − m ≤ n + 1. Thus for fixed n and all but finitely many p , thiscan always be arranged by taking q suitably large.The overall organization of the paper is as follows: Section 2 contains preliminaryresults, including detailed information on the Sylow structure and Sylow normalizersof simple groups containing strongly closed p -subgroups. The main results are provedin Section 3; Theorem 1.3 is proved first and Theorem 1.2 and its corollaries arederived at the end of this section as consequences of it. Section 4 provides interesting TRONGLY CLOSED SUBGROUPS 7 examples of groups, G , possessing strongly closed subgroups, A ; and with an eyeto applications in [FlFo08] we also describe N G ( A ) and N G ( S ) for these cases of G .More explicitly, we describe these first for G simple, and then for split extensions,and finally for certain nonsplit extensions of simple groups. The latter are veryilluminating in the sense that they give an alluring glimpse of what “should be” the B Z /p -cellularization of more general objects.2. Preliminary Results
The special case when A has order p has already been treated in [GLSv3, Proposi-tion 7.8.2]. It is convenient to quote this special case, although with extra effort ourarguments could be reworded to independently subsume it. Proposition 2.1. 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 -subgroupsof G are cyclic, or G is isomorphic to U ( p ) or one of the simple groups listed inconclusions (iii) and (iv) of Theorem 1.2. The authors of this result remark that an immediate consequence of this is theodd-prime version of Glauberman’s celebrated Z ∗ -Theorem. Proposition 2.2.
If an element of odd prime order p in any finite group X does notcommute with any of its distinct conjugates then it lies in Z ( X/O p ′ ( X )) . We record some basic facts about strongly closed subgroups (the second of whichrelies on the odd-prime Z ∗ -Theorem). Lemma 2.3.
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 -subgroupof G/N . (2) If A normalizes a subgroup H of G with O p ′ ( H ) = 1 and A ∩ H = 1 then A centralizes H .Proof. In part (1) let A ≤ S ∈ Syl p ( G ). This result follows immediately from thedefinition of strongly closed applied in the Sylow p -subgroup SN/N of G/N togetherwith Sylow’s Theorem. The proof of (2) is the same as for p = 2 since, as notedearlier, the Z ∗ -Theorem holds also for odd primes: by induction reduce to the casewhere G = AH and C A ( H ) = 1. Then any element of order p in A is isolated, hencelies in the center. (cid:3) RAM ´ON J. FLORES AND RICHARD M. FOOTE
The next few results gather facts about the simple groups appearing in the conclu-sions to Theorems 1.1 and 1.2.The cross-characteristic Sylow structures of the simple groups of Lie type are beau-tifully described in [GL83, Section 10] and reprised in [GLSv3, Section 4.10]. Let L ( q )denote a universal Chevalley group or twisted variation over the field F q . (In the no-tation of [GLSv3], L ( q ) = d L ( q ), where d = 1 , , W denote theWeyl group of the untwisted group corresponding to L ( q ). Except for some smallorder exceptions, L ( q ) is a quasisimple group; for example A ℓ ( q ) ∼ = SL ℓ +1 ( q ) and A ℓ ( q ) ∼ = SU ℓ +1 ( q ). There is a set O ( L ( q )) of positive integers, and “multiplicities” r m for each m ∈ O ( L ( q )), such that |L ( q ) | = q N Y m ∈O ( L ( q )) ( Φ m ( q )) r m where Φ m ( x ) is the cyclotomic polynomial for the m th roots of unity.Let p be an odd prime not dividing q and assume S is a nontrivial Sylow p -subgroupof L ( q ). Let m be the smallest element of O ( L ( q )) such that p (cid:12)(cid:12) Φ m ( q ). Let(2.1) W = { m ∈ O ( L ( q )) | m = p a m , a ≥ } and b = X m ∈W r m where b = 0 if W = ∅ . The main structure theorem is as follows. Proposition 2.4.
Under the above notation the following hold: (1) m is the multiplicative order of q (mod p ) . (2) Except in the case where L ( q ) = D ( q ) with p = 3 , S has a nontrivial normalhomocyclic subgroup, S T , of rank r m and exponent | Φ m ( q ) | p . (3) With the same exception as in (2), S is a split extension of S T by a (possiblytrivial) subgroup S W of order p b (where b is defined in (2.1)), and S W isisomorphic to a subgroup of W . In particular, if p (cid:12)(cid:12) | W | or if pm (cid:12)(cid:12) m forall m ∈ O ( L ( q )) , then S = S T is homocyclic abelian. (4) If L ( q ) = D ( q ) with p = 3 and | q − | = 3 a , then S is a split extension ofan abelian group of type (3 a +1 , a ) by a group of order 3, and S has rank 2. (5) If L ( q ) is a classical group (linear, unitary, symplectic or orthogonal) thenevery element of order p is conjugate to some element of S T . (6) Except in D ( q ) (where S W is not defined), S W acts faithfully on S T ; and inthe simple group L ( q ) /Z ( L ( q )) = L ( q ) we have S W ∼ = S W acts faithfully on TRONGLY CLOSED SUBGROUPS 9 S T except when p = 3 with L ( q ) ∼ = SL ( q ) (with (cid:12)(cid:12) q − but (cid:12)(cid:12) q − ) or SU ( q ) (with (cid:12)(cid:12) q + 1 but (cid:12)(cid:12) q + 1 ). (7) If a Sylow p -subgroup of the simple group L ( q ) /Z ( L ( q )) is abelian but notelementary abelian then p does not divide the order of the Schur multiplier 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 [GLSv3,Table 6.12] we must have L ( q ) of type SL n ( q ), SU n ( q ), E ( q ) or E ( q ) with p dividing( n, q − n, q + 1), (3 , q −
1) or (3 , q + 1) respectively. It follows easily from (6) thatin each of the corresponding simple groups a Sylow p -subgroup cannot be abelian ofexponent ≥ p . (cid:3) We shall frequently adopt the efficient shorthand from the sources just cited forthe latter families.
Notation.
Denote SL n ( q ) by SL + n ( q ) and SU n ( q ) by SL − n ( q ) (likewise for the generallinear and projective groups); and say a group is of type SL ǫn ( q ) according to whether p (cid:12)(cid:12) q − ǫ for ǫ = +1, − ± E ( q ) = E +6 ( q ) and E ( q ) = E − ( q ).The following general result is especially important for the groups of Lie type. Proposition 2.5. If G is any simple group with an abelian Sylow p -subgroup S forany prime p , then N G ( S ) acts irreducibly and nontrivially on Ω ( S ) , and so S ishomocyclic. In particular, a nontrivial subgroup of S is strongly closed if and only ifit is homocyclic of the same rank as S .Proof. See [GLSv3, Proposition 7.8.1] and [GL83, 12-1]. (cid:3)
Proposition 2.6.
Let G be a simple group of Lie type over F q and let p be an oddprime not dividing q . Assume a Sylow p -subgroup S of G is abelian and let A = Ω ( S ) .Then N G ( A ) = N G ( S ) .Proof. The result is trivial if S = A so assume this is not the case; in particularthe exponent of S is at least p . By part (7) of Proposition 2.4, p does not dividethe order of the Schur multiplier of G , so we may assume G is the (quasisimple)universal cover of the simple group. Clearly N G ( S ) ≤ N G ( A ). Moreover, since S ∈ Syl p ( C G ( A )), by Frattini’s Argument N G ( A ) = C G ( A ) N G ( S ). Thus it suffices toshow C G ( A ) = C G ( S ). Since C G ( A ) has an abelian Sylow p -subgroup and since anynontrivial p ′ -automorphism of S must act nontrivially on A , by Burnside’s Theorem C G ( A ) has a normal p -complement. Let ∆ = [ O p ′ ( C G ( A )) , S ]. It suffices to prove S centralizes ∆.Let G be the simply connected universal algebraic group over the algebraic closureof F q , and let σ be a Steinberg endomorphism whose fixed points equal G . In thenotation of Proposition 2.4, since S = S T , by the proof of [GLSv3, Theorem 4.10.2]there is a σ -stable maximal torus T of G containing S . Let C denote the connectedcomponent of C G ( A ), so C is also σ -stable. Note that T ≤ C and since ∆ is generatedby conjugates of S , so too ∆ ≤ C . We may now follow the basic ideas in the proofof [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 component(possibly trivial), and Z ∩ L is a finite group. Since ∆ ≤ C ′ we have ∆ ≤ L . Thegroup of fixed points of σ on L is a commuting product L · · · L n of (possibly solvable)groups of Lie type over the same characteristic as G and smaller rank, and S inducesinner or diagonal automorphisms on each L i . Since ∆ ≤ O p ′ ( C G ( A )) we have∆ ≤ O p ′ ( L · · · L n ) = O p ′ ( L ) · · · O p ′ ( L n ) . If L i is a p ′ -group, then Inndiag( L i ) is also a p ′ -group and so S centralizes L i . On theother hand, if p divides the order of L i , then O p ′ ( L i ) ≤ Z ( L i ); in this case Inndiag( L i )centralizes Z ( L i ). In all cases S centralizes O p ′ ( L i ), as needed. (cid:3) Proposition 2.7.
Let p be any prime, let G be a simple group containing a stronglyclosed p -subgroup, let S ∈ Syl p ( G ) and let Z = Z ( S ) . (1) Assume G ∼ = U ( q ) with q = p n , or G ∼ = Sz ( q ) with p = 2 and q = 2 n . Then S is a special group of type q or q respectively, and N G ( S ) = N G ( Z ) = SH ,where the Cartan subgroup H is cyclic of order ( q − / (3 , q + 1) or q − respectively. In both families H acts irreducibly on both Z and S/Z , and Z isthe unique nontrivial, proper strongly closed subgroup of S . (2) Assume G ∼ = Re ( q ) with p = 3 and q = 3 n , n > . Then S is of class 3, Z ∼ = E q and S ′ = Φ ( S ) = Ω ( S ) ∼ = E q . Furthermore, N G ( S ) = N G ( Z ) = SH , TRONGLY CLOSED SUBGROUPS 11 where the Cartan subgroup H is cyclic of order q − and acts irreducibly onall three central series factors: Z and S ′ /Z and S/S ′ . Thus Z and Ω ( S ) arethe only nontrivial proper strongly closed subgroups of S . (3) Assume G ∼ = G ( q ) for some q with ( q,
3) = 1 and p = 3 . Then Z ∼ = Z is theonly nontrivial proper strongly closed subgroup of S . Furthermore, N G ( Z ) ∼ = SL ǫ ( q ) · according to whether (cid:12)(cid:12) q − ǫ . An element of order 2 in N G ( Z ) − C G ( Z ) inverts Z , and N G ( S ) /S ∼ = QD or E according as | S | = 3 or | S | > respectively. No automorphism of G of order 3 normalizes S andcentralizes both S/Z and a ′ -Hall subgroup of N G ( S ) . (4) Assume G is isomorphic to one of the sporadic groups: J (with p = 3 ); Co , Co , HS , M c (with p = 5 ); or J (with p = 11 ). In each case S isnon-abelian of order p and exponent p , and Z is the only nontrivial properstrongly closed subgroup of S . The normalizer of Z [in G ] is: P GL (9) [in J ], ((4 ∗ SL (3)) · [in Co ], ((4 Y S ) · [in Co ], (8 · [in HS ], (5 · · [in M c ], or (11 · SL (3)) · [in J ]. In G = J we have N G ( S ) /S ∼ = Z ; and in all other cases N G ( S ) = N G ( A ) . (5) Assume G ∼ = J with p = 3 . Then Z ∼ = E and Ω ( S ) ∼ = E are the only non-trivial proper strongly closed subgroups of S . Furthermore, N G ( Z ) = N G ( S ) = SH where H ∼ = Z acts fixed point freely on Ω ( S ) and irreducibly on Z .Proof. Part (1) may be found in [HKS72] and [Suz62]. Part (2) appears in [Wa66].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 = C G ( Z ) ∼ = SL ǫ ( q )according to the condition 3 (cid:12)(cid:12) q − ǫ . The same reference shows G has two conjugacyclasses of elements of order 3: the two nontrivial elements of Z are in one class, andall elements of order 3 in S − Z lie in the other. Now S ≤ SL ǫ ( q ) acts absolutelyirreducibly on its natural 3-dimensional module over F q (or F q in the unitary case),hence by Schur’s Lemma the centralizer of S in C consists of scalar matrices. Thus Z = C C ( S ) = C G ( S ). Since the two nontrivial elements of Z are conjugate in G , N G ( Z ) = C h t i where an involution t may be chosen to normalize S and induce agraph (transpose-inverse) automorphism on C . By canonical forms, all non-centralelements of order 3 in SL ǫ ( q ) are conjugate in GL ǫ ( q ) to the same diagonal matrix u = diag( λ, λ − , λ is a primitive cube root of unity, but are also conjugatein SL ǫ ( q ) to u because the outer (diagonal) automorphism group induced by GL ǫ may be represented by diagonal matrices that commute with u . Thus all elements oforder 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 N C ( S/Z ) = N C ( S ) /Z ; hence they are conjugateunder the faithful action of a 3 ′ -Hall subgroup, H , of N C ( S ) on S/Z . This shows | H | ≥
8. Since a 3 ′ -Hall subgroup H of N G ( S ) acts faithfully on S/Z and has order2 | H | , it must be isomorphic to a Sylow 2-subgroup, QD , of GL (3) as claimed.If | S | = 3 a +1 >
27 then we may describe S as the group, S T , of diagonal matrices of3-power order acted upon by a permutation matrix w of order 3 (where h w i = S W ).Then S T ∼ = Z a × Z a is the unique abelian subgroup of S of index 3 (as | Z | = 3), so N C ( S ) normalizes S T . Let H be a 3 ′ -Hall subgroup of N C ( S ). One easily sees that H must act faithfully on Ω ( S T ) (and centralize Z ), hence | H | ≤
2. Since there isa permutation matrix of order 2 in C normalizing S , | H | = 2. Thus N G ( S ) /S hasorder 4, and is seen to be a fourgroup by its action on Ω ( S T ).To see that Z is the unique nontrivial strongly closed subgroup that is proper in S suppose B is another, so that Z < B . If B contains an element of order 9 — hence anelement 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 areconjugate in C this shows Ω ( S ) ≤ B . It is an exercise that Ω ( S ) = S (the detailsappear 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 andcentralizes S/Z . Then | S : C S ( f ) | ≤ f cannot be a field automorphism as | G ( r ) : G ( r ) | ≥ for all r prime to 3. Thus f must induce an inner automor-phism on G , hence act as an element of order 3 in S T . We have already seen that nosuch element centralizes a 3 ′ -Hall subgroup of N G ( S ), a contradiction. This completesall parts of the proof. (cid:3) Corollary 2.8.
Let p be any prime, let L be a finite simple group possessing a stronglyclosed p -subgroup A that is properly contained in the Sylow p -subgroup S of L . Assumefurther that L is isomorphic to one of the groups L i in the conclusion of Theorem 1.1or Theorem 1.2. Then one of the following holds: (1) N L ( S ) = N L ( A ) , (2) | A | = 3 and L ∼ = G ( q ) for some q with ( q,
3) = 1 , or (3) | A | = 3 , L ∼ = J and N L ( A ) ∼ = 3 P GL (9) . TRONGLY CLOSED SUBGROUPS 13
Proof.
This is immediate from Propositions 2.6 and 2.7. (cid:3) The Proofs of the Main Theorems
In this section we first prove Theorem 1.3; Theorem 1.2 and its corollaries are thenderived from it at the end of this section.3.1.
The Proof of Theorem 1.3.
Throughout this subsection p is an odd prime, G is a minimal counterexample toTheorem 1.3, and A is a nontrivial strongly closed subgroup of G that is a propersubgroup of the Sylow p -subgroup S of G . The minimality implies that if H is anyproper section of G containing a nontrivial minimal strongly closed (with respect to H ) p -subgroup A , then either A is a Sylow subgroup of its normal closure in H or the normal closure of A in H is a direct product of isomorphic simple groups,as described in the conclusion of Theorem 1.2, where overbars denote passage to H/ O A ( H ). In particular, A does not even have to be a subgroup of A , although forthe most part we will be applying this inductive assumption to subgroups A ≤ A ∩ H (which we often show is nontrivial by invoking part (2) of Lemma 2.3).Familiar facts about the families of simple groups, including the sporadic groups,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 results that arecrucial to our arguments. Lemma 3.1. G is a simple group.Proof. Since strong closure inherits to quotient groups, if O A ( G ) = 1 we may applyinduction to G/ O A ( G ) and see that the asserted conclusion holds. Thus we mayassume O A ( G ) = 1, i.e.,(3.1) A ∩ N is not a Sylow p -subgroup of N for any nontrivial N E G .In particular,(3.2) O p ′ ( G ) = 1 . Let G = h A G i and assume G = G . By (3.1), A is not a Sylow p -subgroup of G . Let 1 = A ≤ A be a minimal strongly closed subgroup of G . By the inductivehypothesis A is contained in a semisimple normal subgroup N of G satisfying the conclusions of the theorem. Since N E G it follows that M = h N G i is a semisimplenormal subgroup of G whose simple components are described by Theorem 1.2. Since A is minimal strongly closed in G and 1 = A ≤ A ∩ M , A ≤ M and the conclusionof Theorem 1.3 is seen to hold. Thus(3.3) G is generated by the conjugates of A .By strong closure A ∩ O p ( G ) E G , hence by (3.1), A ∩ O p ( G ) = 1. Thus [ A, O p ( G )] ≤ A ∩ O p ( G ) = 1, i.e., A centralizes O p ( G ). Since G is generated by conjugates of A ,(3.4) O p ( G ) ≤ Z ( G ) . By (3.2) and (3.4), F ∗ ( G ) is a product of commuting quasisimple components, L , . . . , L r ,each of which has a nontrivial Sylow p -subgroup. Since A acts faithfully on F ∗ ( G ),by Lemma 2.3 A ∩ F ∗ ( G ) = 1. The minimality of A then forces A ≤ F ∗ ( G ). Thus A normalizes each L i , whence so does G by (3.3). Now A acts nontrivially on onecomponent, say L , so again by Lemma 2.3, A ∩ L = 1. By minimality of A weobtain A ≤ L E G , so by (3.3) G = L is quasisimple (with center of order a power of p ).Finally, assume Z ( G ) = 1 and let e G = G/Z ( G ). Since A = S but A ∩ Z ( G ) = 1,by Gasch¨utz’s Theorem we must have that S = AZ ( G ) and so e A is strongly closedbut not Sylow in the simple group e G . Since | e G | < | G | , the pair ( e G, e A ) satisfy theconclusions of Theorem 1.3; in particular, e A = Ω ( Z ( e S )) in all cases. If e G is a group ofLie type in conclusion (i), again by Gasch¨utz’s Theorem together with the irreducibleaction of N e G ( e S ) on Ω ( e S ), e A must lift to a non-abelian group in G . In this situation Z ( G ) ≤ A ′ , contrary to A ∩ Z ( G ) = 1. In conclusions (ii), (iii) and (iv) the p -part ofthe multipliers of the simple groups are all trivial, so Z ( G ) = 1 in these cases. In case(v) when e G ∼ = J and e A = Z ( e S ) by the fixed point free action of an element of order8 in N G ( S ) on S it again follows easily that ˜ A must lift to the non-abelian group oforder 27 and exponent 3 in G , contrary to A ∩ Z ( G ) = 1. This shows Z ( G ) = 1 andso G is simple. The proof is complete. (cid:3) Lemma 3.2. A is not cyclic and S is non-abelian.Proof. If A is cyclic then since Ω ( A ) is also strongly closed, the minimality of A gives that | A | = p . Then G is not a counterexample by Proposition 2.1. Likewiseif S is abelian, by Proposition 2.5 it is homocyclic with N G ( S ) acting irreducibly TRONGLY CLOSED SUBGROUPS 15 and nontrivially on Ω ( S ). By minimality of A we must then have A = Ω ( S ) andthe exponent of S is greater than p . None of the sporadic or alternating groups orgroups of Lie type in characteristic p contain such Sylow p -subgroups, so G mustbe a group of Lie type in characteristic = p . Again, G is not a counterexample, acontradiction. (cid:3) Note that because A is a noncyclic normal subgroup of S and p is odd, A containsan abelian subgroup U of type ( p, p ) with U E S . Furthermore, | S : C S ( 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 aspossibilities for the minimal counterexample. The argument used to eliminate thealternating groups is a prototype for the more complicated situation of Lie typegroups, so slightly more expository detail is included. Lemma 3.3. G is not an alternating group.Proof. Assume G ∼ = A n for some n . Since S is non-abelian, n ≥ p . If p (cid:12)(cid:12) n then S is contained in a subgroup isomorphic to A n − , which contradicts the minimality of G (no alternating group satisfies the conclusions in Theorem 1.2). Thus n = ps forsome s ∈ N with s ≥ p .Let E be a subgroup of S be generated by s commuting p -cycles. Since E containsa conjugate of every element of order p in G , A ∩ E = 1. We claim E ≤ A . Let z = z · · · z r ∈ A ∩ E be a product of commuting p -cycles z i in E with r minimal. If r ≥ σ ∈ A n that inverts both z r and z r − and centralizes allother z i ; and if r = 2, since n ≥ r there is an element σ ∈ A n that inverts z andcentralizes z . In either case, by strong closure z σ ∈ A ∩ E and zz σ = z · · · z r − or z respectively. Hence zz σ is an element of A ∩ E that is a product of fewer commuting p -cycles, a contradiction. This shows A contains a p -cycle, hence by strong closure E ≤ A . Now A n contains a subgroup H with(3.5) S ≤ H = N A n ( E ) and H ∼ = Z p ≀ A s . By our inductive assumption H contains a normal subgroup N = O A ( H ) with E ≤ N such that A ∩ N is a Sylow p -subgroup of N and H/N a product of simple componentsdescribed in Theorem 1.2. 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 = E . Since H/E ∼ = A s is not one of the simple groups in Theorem 1.2 it follows that N = H (in the cases where s = 3 or 4 as well), contrary to A = S . This contradiction establishes thelemma.Alternatively, one could argue from (3.5) and induction that S = Ω ( S ), and soagain S = A by strong closure, a contradiction. (cid:3) Lemma 3.4. G is not a classical group (linear, unitary, symplectic, orthogonal) over F q , 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 ≥ F q in the unitary case). The tables in [KL90, Chapter 4] are helpful referencesin this proof.First consider when G is neither a linear group with p dividing q − p dividing q + 1. This restriction implies that p (cid:12)(cid:12) | X : X ′ | and thereis a surjective homomorphism X ′ → G whose kernel is a p ′ -group. Thus we maydo calculations in X in place of G (taking care that conjugations are done in X ′ ).Proposition 2.4 is realized explicitly in this case as follows: There is a decomposition V = V ⊥ V ⊥ · · · ⊥ V s of V ( ⊥ denotes direct sum in the linear case), where Isom( V ) is a p ′ -group, the cyclicgroup of order p has an orthogonally indecomposable representation on each other V i ,the V i are all isometric, and a Sylow p -subgroup of Isom( V i ) is cyclic. Furthermore, X ′ contains a subgroup isomorphic to A s permuting V , . . . , V s and the stabilizer in X ′ of the set { V , . . . , V s } contains a Sylow p -subgroup of X . In other words, we mayassume(3.6) S ≤ H ∼ = Isom( V ) ≀ A s . In the notation of Proposition 2.4, let S ∩ Isom( V i ) = h u i i , where u i acts trivially on V j for all j = i . Then S T = h u , . . . , u s i and S W is a Sylow p -subgroup of A s . Since S is non-abelian, S W = 1 and so s ≥ p ≥
3. Let z i be an element of order p in h u i i ,and let E = h z , . . . , z s i = Ω ( S T ) ∼ = E p s . The faithful action of S W on S T forces Z ( S ) ≤ S T , so A ∩ E = 1. TRONGLY CLOSED SUBGROUPS 17
We claim E ≤ A . As in the alternating group case, let z be a nontrivial elementin A ∩ E belonging to the span of r of the basis elements z i in E with r minimal.After renumbering and replacing each z i by another generator for h z i i if necessary,we may assume z = z · · · z r . If r ≥ σ ∈ G that acts triviallyon z , . . . , z r − and normalizes but does not centralize h z r − , z r i ; and if r = 2, since s ≥ σ ∈ G that centralizes z and normalizes but does notcentralize h z i . In both cases z σ z − is a nontrivial element of A ∩ E that is a productof fewer basis elements. This shows z i ∈ A for some i and so E ≤ A since all z j areconjugate in G .By Proposition 2.4(5) in this setting, every element of order p in G is conjugateto an element of E . Since the extension in (3.6) is split, A S T . By the overallinduction hypothesis applied in H (or because a Sylow p -subgroup of A s is generatedby elements of order p ), it follows that A covers S/S T . We may therefore choose anumbering so that for some x ∈ A , u x = u . Thus u = u u − = [ u , x ] ∈ A ∩ Isom( V ⊥ · · · ⊥ V s − ) . Let Y = G ∩ Isom( V ⊥ · · · ⊥ V s − ) so that Y is also a classical group of the sametype as G over F q . 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 , byinduction applied using a minimal strongly closed subgroup A of A ∩ Y in Y weobtain the following: either A (hence also A ) contains a Sylow p -subgroup of Y , orthe Sylow p -subgroups of Y are homocyclic abelian with A ∩ Y elementary abelianof the same rank as a Sylow p -subgroup of Y . Furthermore, in the latter case aSylow p -normalizer acts irreducibly on A , and hence the strongly closed subgroup A ∩ Y is also homocyclic abelian. Since A ∩ Y contains the element u of order d ,where d = | u | , in either case A ∩ Y contains all elements of order d in S ∩ Y . Since u ∈ S ∩ Y this proves u ∈ A . By (3.6) all u i are conjugate in G to u , 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 divides q − q + 1 respectively (denoted as usual by p (cid:12)(cid:12) q − ǫ ). Now replace the simplegroup G by its universal quasisimple covering SL ǫ ( V ). Likewise replace A by the p -part of its preimage. Thus A is a noncyclic (hence noncentral) strongly closed p -subgroup of SL ǫ ( V ). In this situation S = S T S W where we may assume S T is thegroup of p -power order diagonal matrices of determinant 1 (over F q in the unitary case), and S W is a Sylow p -subgroup of the Weyl group W of permutation matricespermuting the diagonal entries. Furthermore, S T is homocyclic of exponent d , where d = | q − ǫ | p , and is a trace 0 submodule of the natural permutation module for W of exponent d and rank m = dim V . Since A is noncyclic, it contains a noncentralelement z of order p ; and by Proposition 2.4, z is conjugate to an element of S T , i.e.,is diagonalizable. Arguing as above with E = Ω ( S T ) we reduce to the case where z is represented by the matrix diag( ζ , ζ − , , . . . ,
1) for some primitive p th root of unity ζ . The action of W again forces E ≤ A . Again, every element of order p in S isconjugate in G to an element of E , so by strong closure(3.7) Ω ( S ) ≤ A. Consider first when m ≥
5. Then C G ( z ) contains a quasisimple component L ∼ = SL ǫm − ( q ) ′ . Since L contains a conjugate of z , the inductive argument used in thegeneral case shows that A ∩ L contains a diagonal matrix element of order d , hencecontains such an element centralizing an n − A then again yields S T ≤ A ; and as before by induction or because S = S T Ω ( S ) we get A = S , a contradiction.Thus dim V ≤
4, and since S W = 1 we must have p = 3. If G ∼ = SL ǫ ( q ) then let z be represented by the diagonal matrix diag( ζ , ζ , ζ , ζ is a primitive 3 rd rootof unity. Then C G ( z ) contains a Sylow 3-subgroup of G and a component of type SL ǫ ( q ), so the preceding argument leads to a contradiction.Finally, consider when G ∼ = SL ǫ ( q ). The Sylow 3-subgroups of SL ǫ ( q ) are describedin the proof of Proposition 2.7. In both instances S T is homocyclic of rank 2 andexponent d with generators u , u , and with S W = h w i ∼ = Z acting by u w = u and u w = u − u − . Thus u w has order 3, and so u = ( u w ) w − ∈ Ω ( S ). By (3.7), this again forces A = S , which gives the final contradiction. (cid:3) Lemma 3.5. G is not an exceptional group of Lie type (twisted or untwisted) over F q , where q is a prime power not divisible by p .Proof. Assume G = L ( q ) is an exceptional group of Lie type over F q with p (cid:12)(cid:12) q .Throughout this proof we rely on the Sylow structure for G as described in Propo-sition 2.4. It shows, in particular, that we need only consider when the odd prime p TRONGLY CLOSED SUBGROUPS 19
Table 3A
Group Prime p Permissible ( m , r m , p b ) D ( q ) 3 (1 , , ), (2 , , ) G ( q ) 3 (1 , , , , F ( q ) 3 (1 , , ), (2 , , ) F (2 n ) ′ , , E ( q ) 3 (1 , , ), (2 , , )5 (1 , , E ( q ) 3 (1 , , ), (2 , , )5 (2 , , E ( q ) 3 (1 , , ), (2 , , )5 (1 , , , , , , , , E ( q ) 3 (1 , , ), (2 , , )5 (1 , , ), (2 , , ), (4 , , , , , , G and pm (cid:12)(cid:12) m for some m ∈ O ( G ); in all other cases the proposition gives that the Sylow p -subgroup is homocyclic abelian. The cyclotomic factors Φ m ( q ) and their “multiplic-ities” r m for each of the exceptional groups are listed explicitly in [GL83, Table 10:2].Note that 3 (cid:12)(cid:12) q −
1, so in this case m is 1 or 2; also, 5 (cid:12)(cid:12) q −
1, so in this case m is 1, 2, or 4; finally, 7 (cid:12)(cid:12) q −
1, so in this case m is 1, 2, 3, or 6. In the notation ofProposition 2.4, except in the case D ( q ) we have S = S T S W (split extension) where S T is a normal homocyclic abelian subgroup of exponent | Φ m ( q ) | p and rank r m , and | S W | = p b , where b is defined in (2.1).The exceptional groups are listed in Table 3A along with p dividing the order ofthe Weyl group, permissible m such that m = p a m for some m ∈ O ( G ) with a ≥ r m and p b for each of these (in the case of D ( q ) we define3 b so that | S | = ( | Φ m ( q ) | p ) r m b ). We consider all these cases, working from largest to smallest — the latter requiringmore delicate examination. Table 4-1 in [GL83] is used frequently without specificcitation: it lists all the “large” subgroups of various families of Lie type groups thatwe shall employ. It is helpful to keep in mind the description of the order of a Sylow p -subgroup in Proposition 2.4 when comparing the p -part of | G | to that of its Lie-typesubgroups. Case p = 7 p = 7 p = 7: E ( q ) contains both A ( q ) and A ( q ) and so, by inspection of orders,shares a Sylow 7-subgroup with it in the cases (1,8,7) and (2,8,7) respectively (theSylow 7-subgroup order is seen to be 7 · | q − ǫ | for each group). Likewise E ( q )contains both A ( q ) and A ( 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 eliminated. Case p = 5 p = 5 p = 5: The same containments in the preceding paragraph for E ( q ) show thesegroups share a Sylow 5-subgroup in cases (1,7,5) and (2,7,5). Similarly, E ( q ) contains SU ( q ) and shares a Sylow 5-subgroup with it in the case (4,4,5). By minimalitythese p = 5 cases are eliminated.Assume G ∼ = E ( q ). Using the same large subgroups as in the p = 7 case, theSylow 5-subgroup S has a subgroup S of index 5 that lies in a subgroup G of G of type A ( q ) or A ( q ) according to whether we are in cases (1 , , ) or (2 , , )respectively. By Proposition 2.4 applied to G it follows that S is non-abelian; andsince | A | > A ∩ S = 1. Thus by induction applied to a minimal strongly closedsubgroup A ≤ A ∩ S in G we obtain S ≤ A . Moreover, by Proposition 2.4 itfollows that S T ≤ S . Since A is non-abelian and since the normalizer of a Sylow5-subgroup of the Weyl group of E acts irreducibly on the Sylow 5-subgroup of W (which is abelian of type (5,5)), the strongly closed subgroup A containing S T cannothave index 5 in S , a contradiction. This eliminates all E ( q ) cases for p = 5.Adopting the notation following Proposition 2.4, assume G ∼ = E ǫ ( q ), where 5 (cid:12)(cid:12) q − ǫ and S T has rank 6 and index 5 in S . Then G shares the Sylow 5-subgroup S with G = L ∗ L , where L and L are central quotients of SL ǫ ( q ) and SL ǫ ( q ) respectively(both of whose centers have order prime to 5). Since A is not cyclic, it does notcentralize L ; hence it follows from Lemma 2.3 that A ∩ L = 1. Since S ∩ L isnon-abelian, by induction S ∩ L ≤ A . In particular, A contains a homocyclic abeliansubgroup of rank 5 and exponent | q − ǫ | , and S/A is cyclic. Now G also containsa subgroup G = K ∗ K ∗ K with each K i ∼ = SL ǫ ( q ), where we may assume TRONGLY CLOSED SUBGROUPS 21 S ∩ G ∈ Syl ( G ). Each K i contains a homocyclic abelian subgroup B i of rank 2and exponent | q − ǫ | with N K i ( B i ) acting irreducibly on Ω ( B i ). Because S/A iscyclic it follows that B × B × B = S T ≤ 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 type G ( q ) and D ( q ) until the very end. Case p = 3 p = 3 p = 3 and m = 1 m = 1 m = 1: Here 3 (cid:12)(cid:12) q −
1. If G ∼ = F ( q ) then it contains the universalgroup G = B ( q ) u . By inspection of the order formulas, G may be chosen to containa subgroup S of index 3 in S which, by Proposition 2.4, is non-abelian. Since | A | > S ∩ A = 1 so, as usual, the minimality of G forces S ≤ A . Thus S = A has index 3 in S . Furthermore, since a Sylow 3-subgroup of the Weyl group of B has order 3, we get that A has an abelian subgroup of index 3. But now by [GLSv3,Table 4.7.3A] there is an element t of order 3 in G such that C = O ′ ( C G ( t )) = L ∗ L where L i ∼ = SL ( q ) for i = 1 ,
2. Choose a suitable representative of this class so that C S ( t ) ∈ Syl ( C ). Then A ∩ L i Z ( L i ), so because each Sylow subgroup S ∩ L i isnon-abelian, by induction S ∩ L i ≤ A for i = 1 ,
2. This gives a contradiction because S ∩ L L clearly does not have an abelian subgroup of index 3.Since E ( q ) shares a Sylow 3-subgroup with a subgroup of type F ( q ) this familyis eliminated by minimality of G .Consider when G is one of E ( q ), E ( q ) or E ( q ). In these cases S T is homocyclicof the same rank as G and S T lies in a maximal split torus T of G with W = N G ( T ) /C G ( T ) isomorphic to the Weyl group of G . Note that W acts on the Sylow 3-subgroup S T of T ; moreover, in each case W acts irreducibly on Ω ( S T ), and Z ( S ) ≤ S T . By strong closure of A we obtain(3.8) Ω ( S T ) ≤ A. There are containments: F ( q ) ≤ E ( q ) ≤ E ( q ) ≤ E ( q ), with corresponding con-tainments of their maximal split tori. Thus by (3.8), in each exceptional family A nontrivially intersects a subgroup, G , of G of smaller rank in this chain. Since theSylow 3-subgroups of each G are non-abelian, by minimality of G and the precedingresults we get that A contains a Sylow 3-subgroup of the respective subgroup G .Since then A is non-abelian, it is not contained in S T . Now the Weyl group of G isof type U (2) · Z × S (2), or 2 · O +8 (2) ·
2, so by induction applied in N G ( T ) it follows that A covers a Sylow 3-subgroup of W . Finally, the irreducible action of W on S T /Φ ( S T ) forces S T ≤ A , and so A = S , a contradiction. Case p = 3 p = 3 p = 3 and m = 2 m = 2 m = 2: Here 3 (cid:12)(cid:12) q + 1. The argument employed when 3 (cid:12)(cid:12) q − F ( q ) as a possibility (using L i ∼ = SU ( q ) in this case).The groups F (2 n ) ′ — including the Tits simple group — share a Sylow 3-subgroupwith their subgroups SU (2 n ), and so are eliminated by induction. Also, E ( q ) sharesa Sylow 3-subgroup with its subgroup F ( q ), hence it is eliminated. To eliminate E ( q ), E ( q ) and E ( q ) we refer to the table of centralizers of elements of order 3 inthese groups: [GLSv3, Table 4.7.3A].First assume G ∼ = E ( q ). By [GLSv3, Table 4.7.3B], G contains a subgroup X ∼ = L × L , where the two components are conjugate and of type U ( q ). We may assume S ∩ X ∈ Syl ( X ). Since Ω ( S T ) is the unique elementary abelian subgroup of S ofrank 8, Ω ( S T ) ≤ X ; in particular, A ∩ X = 1. As usual, by minimality of G weobtain S ∩ X ≤ A , and the “toral subgroup” for S ∩ X lies in S T . Order considerationsthen give S T ≤ A and | S : A | ≤ . Now the centralizer of an element of order 3 in Z ( S ) is of type ( E ( q ) ∗ SU ( q ))3, where the two factors share a common center oforder 3. Since S T ≤ A it follows that A acts nontrivially on, hence contains a Sylow3-subgroup of, each component (or of SU (2) when q = 2). This implies A covers S/S T ∼ = S W , as needed to give the contradiction A = S .Let G ∼ = E ( q ). Then G contains a subgroup X ∼ = SU ( q ) with S ∩ X ∈ Syl ( X ).Since S ∩ X has the same “toral subgroup” as S , as usual we obtain S ∩ X ≤ A , S T ≤ A and | S : A | ≤ . Now S also contains an element of order 3 whosecentralizer has a component of type E ( q ) (universal version). Since as usual A contains a Sylow 3-subgroup of this component it follows that A covers S/S T and so A = S , a contradiction.Finally, assume G ∼ = E ( q ). Since by [CCNPW] E (2) shares a Sylow 3-subgroupwith a subgroup of type F i , 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 ∼ = ( L ∗ L ∗ L )(3 × L i ∼ = SU ( q ), the central product L L L has a center of order 3, anelement of S cycles the three components, and another element of S induces outerdiagonal automorphisms on each L i . As usual, it follows easily that A contains aSylow 3-subgroup of S ∩ X . By order considerations | S T : S T ∩ A | ≤ | S : A | ≤ . TRONGLY CLOSED SUBGROUPS 23
Now there is an element t of order 3 in S such that C = C G ( t ) = D ∗ T , where D ∼ = D − ( q ) and T ∼ = Z q +1 , and we may choose t so that S = C S ( t ) ∈ Syl ( C ). Let S = S ∩ D and S = S ∩ T ,and note that h t i = Ω ( T ). Since the Schur multiplier of D has order prime to 3, S = S × S . It follows as usual that S ≤ A .Now let w ∈ S − S and let t = t w . Then t = t and S ∈ Syl ( C G ( t )). Bysymmetry, the strongly closed subgroup A contains the Sylow 3-subgroup S w of thecomponent D w of C G ( t ). Since t acts faithfully on D , so too S w acts faithfully on D , from which it follows that S ≤ S S w ≤ A. Moreover, A contains the “toral subgroup” of C of type ( q + 1) (in the universalversion of G ), so S T ≤ A and hence A is the subgroup of S that normalizes eachcomponent L i of X . Since S W is generated by elements of order 3 (in the universalversion of G ), S = A h x i for some element x of order 3. Since no conjugate of x lies in A we may further assume C S ( x ) ∈ Syl ( C G ( x )). Since h x i cycles L , L , L it followsthat the 3-rank of C G ( x ) is at most 5: this restricts the possibilities for the type of x in [GLSv3, Table 4.7.3A]. In all possible cases C G ( x ) contains a product, L , of oneor two components with C ( L ) cyclic. The same argument that showed S ≤ A maynow be applied to show x ∈ A , a contradiction. This completes the proof for thesefamilies. Case G ( q ) G ( q ) G ( q ) and D ( q ) D ( q ) D ( q ) where q ≡ ǫ (mod 3) q ≡ ǫ (mod 3) q ≡ ǫ (mod 3): If G ∼ = G ( q ) then by Proposition 2.7 Z ( S ) ∼ = Z is the unique candidate for A , contrary to Lemma 3.2. Thus the minimalcounterexample is not of type G ( q ).Assume G ∼ = D ( q ). Then G contains a subgroup G isomorphic to G ( q ) (thefixed points of a graph automorphism of order 3), and by order considerations wemay assume S = S ∩ G is Sylow in G and so has index 3 in S . As noted above, h z i = Z ( S ) is of order 3 and is the unique nontrivial strongly closed (in G ) propersubgroup of S . Consider first when | A ∩ S | >
3. Then since S is non-abelian,induction applied to G gives S ≤ A , and so A = S . Since by Proposition 2.1, z G ∩ S = { z ± } , whereas h z i is not strongly closed in G , there must be G -conjugatesof z in S − S , contrary to A being strongly closed (one can see this fusion in a subgroupof D ( q ) of type P GL ǫ ( q )). Thus A ∩ S = h z i and so by Lemma 3.2, A = h z i × h y i with z ∼ y in G .Since [ S, y ] ≤ h z i , y centralizes Φ ( S ). Since D ( q ) has 3-rank 2 and y / ∈ Φ ( S ), byProposition 2.4(4) we must have | S | = 3 . But then S is the non-abelian group oforder 27 and exponent 3, and y centralizes a subgroup of index 3 in it, contrary tothe 3-rank of D ( q ) being 2. This eliminates the possibility that G ∼ = D ( q ) and socompletes the consideration of all cases. (cid:3) Lemma 3.6. G is not a group of Lie type (untwisted or twisted) in characteristic p .Proof. Assume G is of Lie type (untwisted or twisted) over F q where q = p n . Since G is a counterexample, it follows from Proposition 2.5 that G has BN -rank ≥
2. Anend-node maximal parabolic subgroup P for each of the Chevalley groups (untwistedor 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 totallyisotropic one-dimensional subspace of the natural module.) For the groups of BN -rank 2 the other maximal parabolic, P , is also described in [GLS93]. In each group P i = Q i L i H , where Q i = O p ( P i ), L i is the component of a Levi factor of P i and H isa p ′ -order Cartan subgroup.Except for the 5-dimensional unitary groups and some groups over F (which willbe dealt with separately), for some i ∈ { , } the group M = O p ′ ( P i ) satisfies thefollowing conditions: Properties 3.1A. (1) S ≤ M ,(2) F ∗ ( M ) = O p ( M ),(3) M = M/O p ( M ) is a quasisimple group of Lie type in characteristic p ,(4) M is not isomorphic to U ( p n ) or Re (3 n ) (when p = 3), for any n ≥ O p ( M ) , M ] = O p ( M ), and(6) if Q = O p ( M ) and Z = Ω ( Z ( S )), then one of the following holds: (i): Q is elementary abelian of order q k for some k , or (ii): Q is special of type q k for some k , all subgroups of order p in Z areconjugate in G , and z g ∈ S − Q for some z ∈ Z , g ∈ G .Basic information about this parabolic is listed in Table 3B. The last column ofTable 3B indicates which of the two alternatives in Properties 3.1A(6) holds. The TRONGLY CLOSED SUBGROUPS 25 proofs that the fusion in Properties 3.1A(6ii) holds in each case may be found in[CKS76].
Table 3B
Group Parabolic
Q L/Z ( L ) 3.1A(6) L k ( q ), k ≥ P q k − L k − ( q ) (i) O ± k ( q ), k ≥ P q k − O ± k − ( q ) (i) S k ( q ), k ≥ P q k − S k − ( q ) (ii) U k ( q ), k ≥ k = 5 P q k − U k − ( q ) (ii) E ( q ) P q L ( q ) (ii) E ( q ) P q O +12 ( q ) (ii) E ( q ) P q E ( q ) (ii) E ( q ) P q U ( q ) (ii) G ( q ), q > P q L ( q ) (ii) F ( q ) P q S ( q ) (ii) D ( q ) P q L ( q ) (ii) U ( q ) P q U ( q ) (ii)Putting aside the last row for the moment, let M = O p ′ ( P i ) be chosen accordingto Table 3B. Since M does not have any composition factors isomorphic to U ( p n ) or Re (3 n ), the minimality of G gives inductively that A ∈ Syl p ( h A M i ). If A Q , thenby the structure of M in Properties 3.1A(3) and (5), M ≤ h A M i . But then A = S by (1), a contradiction. Thus(3.9) A ≤ Q and A E M. Assume first that Properties 3.1A(6ii) holds. Then since A E S , Z ∩ A = 1. Thestrong closure of A together with (6ii) forces Z ≤ A , contrary to the existence ofsome z g ∈ S − Q . This contradiction shows that G can only be among the familiesin 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 thesecases 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 thequotient space. Since the dimension of the space is at least 3, one easily exhibitsnoncommuting transvections that stabilize a common maximal flag; hence there areconjugates of elements of Q in S that lie outside of Q , a contradiction.In U ( q ) for q ≥ P is special of type q with Z = Z ( S ) = Z ( Q ) and all subgroups of order p in Z conjugate in P (so Z ≤ A ).As in the other unitary groups, there exist z ∈ Z and g ∈ G such that z g ∈ S − Q .Now L ∼ = U ( q ) acts irreducibly on Q /Z and, by the strong closure of A , A ∩ Q isnormal in P . Since z g ∈ A and [ Q , z g ] ≤ A ∩ Q , the irreducible action of L forces Q ≤ A . But now there is a root group U of type U ( q ) with U contained in Q suchthat S = Q U x , for some x ∈ 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 notquasisimple: G ∼ = L ( q ), L (3), G (3), S (3), or U ( q ) (in line 3 of Table 3B, S ( q ) = L ( q )). Properties of small order groups may be found in [CCNPW]. The groups L ( q ) have elementary abelian Sylow p -subgroups so G is not a counterexample inthis instance. In L (3) we have S ∼ = 3 and the action of the two maximal parabolicsubgroups (stabilizers of one- and two-dimensional subspaces) easily show that thestrong closure Z ( S ) in S is all of S , contrary to A = S .If G ∼ = G (3) then since G has two (isomorphic) maximal parabolics containing S , A is not normal in one of them, say P . By [CCNPW], P = ( W × U ) : L where W ∼ = 3 , U ∼ = Z × Z , O ( P ) = W U , and L ∼ = GL (3) acts naturally on both U and W/W ′ . Since A projects onto a subgroup of order 3 in P /O ( P ) ∼ = L , we seethat [ A, W ] W ′ and [ A, U ] = 1. Both these commutators lie in the strongly closedsubgroup A , so the action of L forces O ( P ) ≤ A . Thus A = S , a contradiction.If G ∼ = S (3) there are maximal parabolics of type P = 3 : SL (3) and P =3 ( S × Z ). Since P = N G ( Z ( S )) it follows that the S Levi factor in P actsirreducibly on O ( P ). Now A ∩ O ( P ) = 1 so O ( P ) ≤ A . Likewise since A is anoncyclic strongly closed subgroup, it follows easily from the action of the Levi factorin P that O ( P ) ≤ A . These together give A = S , a contradiction.Finally, assume G ∼ = U ( q ). From the isomorphism U ( q ) ∼ = O +6 ( q ) we see that G contains a maximal parabolic P = q O +4 ( q ) ∼ = q L ( q ), where the Levi factoris irreducible on the (elementary abelian) unipotent radical. This case has beeneliminated by previous considerations. This final contradiction completes the proofof the lemma. (cid:3) TRONGLY CLOSED SUBGROUPS 27
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 theirproofs may be found in [Asc94]. Facts from these sources are quoted without fur-ther attribution. Verification that the sporadic groups in conclusions (iv) and (v) ofTheorem 1.2 indeed have strongly closed subgroups as asserted may also be found inthese references. We clearly only need to consider groups where p divides the order;indeed, when the Sylow p -subgroup has order exactly p it is elementary abelian and G is not a counterexample in these cases.If | S | = p , then in all cases the Sylow p -subgroup is non-abelian of exponent p and, with the exception of M , N G ( S ) acts irreducibly on S/Z ( S ). In M with p = 3: S contains distinct subgroups U and U , each of order 9, such that N G ( U i )acts irreducibly on U i for each i . Since A is noncyclic and strongly closed, in all casesthese conditions force A = S , a contradiction. Thus we are reduced to consideringwhen | S | ≥ p .We first argue that the following general configuration cannot occur in G : Properties 3.1B. (1) Z ( S ) = Z ∼ = Z p ,(2) N = N G ( Z ) has Q = O p ( N ) extraspecial of exponent p and width w > Q ∼ = p w ),(3) N acts irreducibly on Q/Q ′ , and(4) N/Q does not have a nontrivial strongly closed p -subgroup that is proper ina Sylow p -subgroup of N/Q .By way of contradiction assume these conditions are satisfied in G . If A Q then by (4) we obtain that A covers a Sylow p -subgroup of N/Q . In this case, theirreducible action of N on Q/Q ′ 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 some x ∈ Q − Z such that x ∼ z for z ∈ Z . Thus by Sylow’s Theorem there is some g ∈ G such that C Q ( x ) g ≤ S and x g = z. By strong closure, C Q ( x ) g ≤ Q . But since Q has width > Z g ≤ ( C Q ( x ) g ) ′ ≤ Q ′ = Z and so g normalizes Z . This contradicts the fact that z g − / ∈ Z and so proves these properties cannot hold in G .Most sporadic groups are eliminated because they satisfy Properties 3.1B, or be-cause they share a Sylow p -subgroup with a group that is eliminated inductively. Allcases where | S | ≥ p are listed in Table 3C along with the isomorphism type of thecorresponding normalizer of a p -central subgroup (or another “large” subgroup, orreason for elimination). Some additional arguments must be made in a few cases.When p = 5 and G ∼ = Co the extraspecial Q = O ( N ) listed in the table haswidth 1. As before, if A Q then the irreducible action of N on Q/Q ′ forces A = S ,a contradiction. Thus A ≤ Q and again the irreducible action yields A = Q . However G contains a subgroup G ∼ = Co whose Sylow 5-subgroup S is isomorphic to Q andhas index 5 in S . Since | A ∩ S | ≥
25, the irreducible action of N G ( S ) on S /S ′ forces S ≤ A , and hence S = A . But by Proposition 2.1, Z ( S ) is strongly closedin G but not strongly closed in G . Thus there is some g ∈ G such that Z ( S ) g ≤ S but Z ( S ) g S . This contradicts the fact that A = S is strongly closed in G , andso G = Co .When p = 3 and G ∼ = F i it contains a subgroup H of type O +8 (3) : S that maytherefore be chosen to contain S . Let H = H ′′ ∼ = O +8 (3). By Lemma 2.3, A ∩ H = 1;and so by induction A contains the non-abelian Sylow 3-subgroup S = S ∩ H of H . Thus | S : A | = 3. Now H is generated by 3-transpositions in G , and so thereare 3-transpositions t, t such that D = h t, t i ∼ = S and H = H : D . Likewise t inverts some element of order 3 in H , i.e., there is some t ∈ H h t i suchthat D = h t, t i ∼ = S . By the rank 3 action of G on its 3-transpositions, D and D are conjugate in G . Thus D ′ is conjugate to the subgroup D ′ of H , contrary to A being strongly closed. This proves G = F i .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 3 A in S has isomorphismtype N = N G ( h A i ) ∼ = ( Z × H ) . H ∼ = G (3) . TRONGLY CLOSED SUBGROUPS 29
Table 3CGroup Z ( S ) normalizer (or other reason) p = 7 p = 7 p = 7 M (3 × S ) p = 5 p = 5 p = 5 Ly ((4 ∗ SL (9)) . Co GL (5) HN (2 (5 · B ((( Q ∗ D ) A ) · M ((4 ∗ J ) · p = 3 p = 3 p = 3 M cL (2 S ) Suz U (3)2 Ly M cL O ’ N (one class of Z and S = Ω ( S )) Co GSp (3) Co (( D ∗ Q ) · S ) Co ((4 ∗ SL (9)) · F i ( S ≤ O (3)) F i ( S ≤ O +8 (3) : S ) F i ′ ( U (2) · HN (4 ∗ SL (5)) T h (see separate argument) B (2 O − (2)) M (2 Suz ) · B in Z ( S ) ∩ A commutes with 3 A and therefore actsnontrivially on H , by induction A contains a Sylow 3-subgroup of H . In the Atlasnotation for characters of G (3), the character χ of degree 248 of T h restricts to Z × H as χ | Z × H = 1 ⊗ ( χ + χ ) + ( ω + ω ) ⊗ χ where the characters of the Z factor are denoted by their values on a generator. Bycomparison of the values of these on the G (3)-classes it follows that H contains arepresentative of every class of elements of order 3 in T h . The calculations in [Wi98]show that S = Ω ( S ), which leads to A = S , a contradiction.This eliminates all sporadic simple groups as potential counterexamples, and socompletes the proof of Theorem 1.3. (cid:3) The Proof of Theorem 1.2.
This subsection derives Theorem 1.2 as a consequence of Theorem 1.3. Throughoutthis subsection G is a minimal counterexample to Theorem 1.2.Since strong closure inherits to quotient groups, if O A ( G ) = 1 we may apply induc-tion to G/ O A ( G ) and see that the asserted conclusion holds. Thus we may assume O A ( G ) = 1, and consequently(3.10) A ∩ N is not a Sylow p -subgroup of N for any nontrivial N E G and O p ′ ( G ) = 1.Likewise if G = h A G i then by Frattini’s Argument, G = G N G ( A ), whence h A G i = h A G i . Thus we may replace G by G to obtain(3.11) G is generated by the conjugates of A .By strong closure A ∩ O p ( G ) E G , whence by (3.10), A ∩ O p ( G ) = 1. Since[ A, O p ( G )] ≤ A ∩ O p ( G ) = 1, by (3.11) we have(3.12) O p ( G ) ≤ Z ( G ) . Consequently F ∗ ( G ) is a product of subnormal quasisimple components L , . . . , L r with O p ′ ( L i ) = 1 for all i . Moreover S i = S ∩ L i is a Sylow p -subgroup of L i and S i = 1 by (3.10).We argue that each component of G is normal in G . By way of contradictionassume { L , . . . , L s } is an orbit of size ≥ G on its components.Let Z = A ∩ Z ( S ), so that Z normalizes each L i . Thus N = ∩ si =1 N G ( L i ) is aproper 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 mindthat components of N are necessarily components of G and O B ( N ) = 1 — andafter possible renumbering, there are simple components L , . . . , L t of N that satisfy TRONGLY CLOSED SUBGROUPS 31 the conclusion of Theorem 1.2 with B ∩ L i = 1, these are all the components of N satisfying the latter condition, and t ≥
1. By Frattini’s Argument G = N G ( B ) N fromwhich it follows that L · · · L t E G . The transitive action of G in turn forces t = s .Thus A permutes { L , . . . , L s } and 1 = A ∩ L i < S i . If A does not normalize one ofthese components, say L ai = L j for some i = j and a ∈ A , then S i S ai = S i × S j . Butthen [ S i , a ] ( A ∩ L i ) × ( A ∩ L j ), contrary to A E S . Thus A must normalize L i for1 ≤ i ≤ s . Since A ≤ N E G , (3.11) gives N = G , a contradiction. This proves(3.13) every component of G is normal in G .The preceding results also show that A acts nontrivially on each L i . By Lemma 2.3, A i = A ∩ L i = 1 and A i is not Sylow in L i for every i . By Theorem 1.3 applied toeach L i using a minimal strongly closed subgroup of A i we obtain(3.14) F ∗ ( G ) = L × L × · · · × L r and each L i is one of the simple groups described in the conclusion of Theorem 1.2.Moreover, in each of conclusions (i) to (v), by Propositions 2.5 and 2.7, A i is asubgroup of L i described in the respective conclusion.It remains to verify that the action of A is as claimed when A F ∗ ( G ). Theautomorphism group of each L i is described in detail in [GLSv3, Theorem 2.5.12 andSection 5.3] — these results are used without further citation.Let S ∗ = S ∩ F ∗ ( G ) = S × · · · × S r , let H ∗ = H × · · · × H r , where H i is a p ′ -Hallcomplement to S i in N L i ( S i ), and let N ∗ = AS ∗ H ∗ . Note that O p ′ ( N ∗ ) = C H ∗ ( S ∗ )is A -invariant. Now in all cases [ A, S i ] ≤ A ∩ S i ≤ Φ ( S i ), that is, A commutes withthe action of H ∗ on S ∗ /Φ ( S ∗ ). This forces A ≤ O p ′ ,p ( N ∗ ). By strong closure of A we get that AO p ′ ( N ∗ ) E N ∗ . Thus N N ∗ ( A ) covers H ∗ /C H ∗ ( S ∗ ). Let H be a p ′ -Hallcomplement to AS ∗ in N N ∗ ( A ); we may assume H ∗ = HC H ∗ ( S ∗ ). We have a Fittingdecomposition(3.15) A = [ A, H ] A F where A F = C A ( H ) . By Propositions 2.5 and 2.7 each A i is abelian and H i , hence also H , acts withoutfixed points on each A i . Since [ A, H ] ≤ A ∩ F ∗ ( G ) we therefore obtain(3.16) [ A, H ] = A × · · · × A r and A F ∩ [ A, H ] = 1 . We now determine the action of A F on L i for each isomorphism type in conclusions(i) to (v). First suppose A F acts trivially on some L i , say for i = 1. In this situation A = A × B where B = ( A × · · · × A r ) A F = A ∩ C G ( L ). Then h A G i = L × h B G i , andso we may proceed inductively to identify h B G i and conclude that Theorem 1.2 isvalid. We now observe that A F acts trivially on all components listed in conclusions(ii) to (v) as follows: If L is one of these cases, it follows from Proposition 2.7 that C H ( S ) = 1 and so A F centralizes a p ′ -Hall subgroup of N L ( S ). In case (ii) ofthe conclusions, if L is a Lie-type simple group in characteristic p and BN -rank 1,by [GL83, 9-1] no automorphism of order p centralizes a Cartan subgroup of L , so A F acts trivially on L . If L ∼ = G ( q ) is described by case (iii) of the conclusion,then since [ S , A F ] ≤ A , the last assertion of Proposition 2.7(3) shows that A F actstrivially on L . And in cases (iv) and (v) of the conclusions, when L is a sporadicgroup, none of the target groups admits an outer automorphism of order p , and noinner automorphism that normalizes a Sylow p -subgroup also commutes with a p ′ -Hallsubgroup of its normalizer. Thus A F acts trivially in these instances too.It remains to consider when every L i is described by conclusion (i): L = L i is agroup of Lie type over the field F q i where p (cid:12)(cid:12) q i and the Sylow p -subgroups are abelianbut not elementary abelian. Since A F commutes with the action of a p ′ -Hall subgroupof N L ( S i ), it follows from Proposition 2.5 that A F induces outer automorphisms on L . The outer diagonal automorphism group of L has order dividing the order of theSchur multiplier of L , so by Proposition 2.4(7) no element of G induces a nontrivialouter diagonal automorphism of p -power order on L . Since Sylow 3-subgroups of D ( q ) and D ( q ) are non-abelian, L does not admit a nontrivial graph or graph-fieldautomorphism when p = 3. This shows A F must act as field automorphisms on L ,and hence A F /C A F ( L ) is cyclic.Now G is generated by the conjugates of A , hence the group e G = G/LC G ( L ) ofouter automorphisms on L is generated by conjugates of e A F . This implies via [GLSv3,Theorem 2.5.12] that(3.17) e G = e D e A F and e D = [ e D, e A F ]where e D is a cyclic p ′ -subgroup of the outer-diagonal automorphism group of L nor-malized by the cyclic p -group e A F of field automorphisms. Moreover, since p > L is of type E ( q ), E ( q ) or D m ( q ), the action of e A F on e D in (3.17) implies that e D is trivial except in the cases where L is a linear or unitary group. TRONGLY CLOSED SUBGROUPS 33 A p ′ -order subgroup D that covers the section e D for every L i may be defined asfollows (even in the presence of L i that are not of type (i)): We have now establishedthat S = S ∗ A F , and that S ∗ is a Sylow p -subgroup of the (normal) subgroup G D of G inducing only inner and diagonal automorphisms on F ∗ ( G ). Thus N G D ( S ∗ )has a p ′ -Hall complement, which is then a complement to S = S ∗ A F in N G ( S ∗ ).Since [ S ∗ , A F ] ≤ Φ ( S ∗ ), A F commutes with the action on S ∗ of this p ′ -Hall subgroup.As e D = [ e D, A F ], any choice for D must lie in C G ( S ∗ ). However, C G ( S ∗ ) has anormal p -complement, so any D must lie in O p ′ ( C G ( S ∗ )). Thus [ O p ′ ( C G ( S ∗ )) , A F ] =[ O p ′ ( C G ( S ∗ )) , S ] covers e D for every component L i (and centralizes all componentsthat are not of type P SL or P SU ).Finally note that in every case A ′ F centralizes L i for every i . Since then A ′ F cen-tralizes F ∗ ( G ), it must be trivial, that is, A F is abelian. Since A F /C A F ( L i ) is cyclicfor all i , it follows that A F = A F / ∩ ri =1 C A F ( L i ) has rank at most r , as asserted. Thiscompletes the proof of Theorem 1.2.3.3. The Proofs of Corollaries 1.4 and 1.5.
Considering both corollaries at once, assume the hypotheses of Corollary 1.4 hold.The result is trivial if either A = S (in which case O A ( G ) = G ) or A = 1 (in whichcase G = 1). By passing to G/ O A ( G ) we may assume O A ( G ) = 1. Since G isgenerated by conjugates of A , Theorems 1.1 and 1.2 imply that(3.18) G = ( L × · · · × L r )( D · A F ) , where the L i , D and A F are described in their conclusions (with both D and A F trivial when p = 2). Let S i = S ∩ L i and A i = A ∩ L i .For each i let Z i be a minimal nontrivial strongly closed subgroup of A ∩ L i , andlet Z = Z × · · · × Z r . Then Z is strongly closed in G , and by Propositions 2.5 and2.7, Z is contained in the center of S . It is immediate from Sylow’s Theorem and theweak closure of Z that N G ( Z ) controls strong G -fusion in S . Now N G ( Z ) = ( N L ( Z ) × · · · × N L r ( Z r ))( D · A F )where by the proof of Theorem 1.2, D = [ D, A F ] may be chosen to be an S -invariant p ′ -subgroup centralizing each S i . This implies(3.19) M = ( N L ( Z ) × · · · × N L r ( Z r )) A F controls strong G -fusion in S . It suffices therefore to show that N M ( A ) controls strong M -fusion in S . Furthermore, N M ( A ) controls strong M -fusion in S if and only if the corresponding fact holds in M/O p ′ ( M ); so we may pass to this quotient and therefore assume O p ′ ( M ) = 1 (with-out encumbering the proof with overbar notation, since all normalizers considered arefor p -groups).If L i is a Lie-type component with S i abelian then, as noted in the proof of Theo-rem 1.2, N L i ( Z i ) = N L i ( S i ) and A F commutes with the action on S i of an A F -stable p ′ -Hall subgroup H i of this normalizer. Since O p ′ ( M ) = 1 it follows that H i actsfaithfully on S i , and so [ A F , H i ] = 1. On the other hand, if L i is not of this type,[ A F , L i ] = 1. Thus (reading modulo O p ′ ( M )) we have(3.20) M = SC M ( A F )and so N M ( A ) = N M ( A ∗ ), where A ∗ = A · · · A r .For every component L i that is not of type G ( q ) or J , by Corollary 2.8, N L i ( Z i ) = N L i ( S i ); and therefore in these components N L i ( Z i ) = N L i ( A i ) too. However, for acomponent L i of type G ( q ) or J (with p = 3), by Proposition 2.7 we must have Z i = A i . In all cases we have N L i ( Z i ) = N L i ( A i ). Hence N M ( A ∗ ) = N M ( Z ) = M and the first assertion of Corollary 1.4 holds by (3.19). This also establishes thesecond assertion unless p = 3 and some components L i are of type G ( q ) or J , wherethe possibility that | S i | > in these exceptions is excluded by the hypotheses ofCorollary 1.4.In the remaining case let S ∗ = S × · · · × S r , where S , . . . , S k are the Sylow 3-subgroups of the components of type G ( q ) or J , and S k +1 , . . . , S r are the remainingones. Again by (3.20), N M ( S ) = N M ( S ∗ ) so we must prove the latter normalizercontrols strong M -fusion in S ; indeed, it suffices to prove control of fusion in S ∗ .Now N M ( S ∗ ) controls strong M -fusion in S ∗ if and only if the corresponding resultholds in each direct factor. This is trivial for i > k as S i is normal in that factor. For1 ≤ i ≤ k the result is true since S i = 3 , i.e., S i has a central series 1 < Z i < S i whose terms are all weakly closed in S i with respect to N L i ( Z i ) (see, for example,[GiSe85]). This establishes the final assertion of Corollary 1.4.In Corollary 1.5 observe that by Theorem 1.2, once O A ( G ) is factored out we haveequation (3.18) holding, and since A F acts without fixed points on the cyclic quotient TRONGLY CLOSED SUBGROUPS 35 D/ ( D ∩ L · · · L r ), we must have N G ( A ) ≤ ( L · · · L r ) A F . Thus by (3.20) we have N G ( A ) = N M ( A ) ≤ SC M ( A F ) O p ′ ( M ) . Since N M ( A ) ∩ O p ′ ( M ) centralizes A we have N G ( A ) ≤ SC M ( A F ), and hence N G ( A ) = SC M ( A F ). All parts of Corollary 1.5 now follow.4. Examples
Throughout this section p is any prime, G is a finite group possessing a nontrivialSylow p -subgroup S . In this section we describe some families of groups possessingstrongly closed subgroups A contained in S . Let A ( S ) denote the unique smalleststrongly closed (with respect to G ) subgroup of S that contains Ω ( S ). We focusprimarily on groups where A = A ( S ) = S , as these groups provide illuminatingexamples of fusion, and control (or failure of control) of fusion in S by N G ( S ); andtherefore we describe N G ( A ) and N G ( S ) in our examples. In particular, in Section 4.3we show that the extra hypotheses in the last sentence of Corollary 1.4 are necessary.Our constructions are also significant to homotopy theory, as they provide interestingexamples of cellularizations of classifying spaces, as detailed in [FlFo08].First of all, an example where both D and A F are nontrivial is when G = P Γ L ( q )with p = 5 and q = 3 . Here the simple group L = P SL ( q ) has an abelian Sylow5-subgroup of type (25,25), P GL ( q ) /L is the cyclic outer diagonal automorphismgroup of L of order 11 (this is DL/L ), and h f i = A F induces a group of order 5of field automorphisms on P GL ( q ); in particular, G/L is the non-abelian groupof order 55. If f ∈ S ∈ Syl ( G ), then A = Ω ( S ) = h f, Ω ( S ∩ L ) i is elementaryabelian of order 5 and strongly closed in S with respect to G , and A ∗ = Ω ( 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 inthe universal group GL ( q ) h f i — also denoted by G — via its action on an 11-dimensional F q -vector space V (since the center of GL ( q ) has order prime to 5) —see the proof of Lemma 3.4 for some general methodology. Let G ∗ = GL ( q ) and S ∗ = S ∩ G ∗ . Then one sees that N G ( A ∗ ) = N G ( S ∗ ) is contained in a subgroup H = (( G × G ) h t i × C ) h f i where G i ∼ = GL ( q ), C ∼ = GL ( q ), t interchanges the two factors and f induces fieldautomorphisms on all three factors and commutes with t (here G × G × C actsnaturally on a direct sum decomposition of V ). Let S i = S ∩ G i , so S i is cyclic oforder 25 and acts F q -irreducibly on the 4-dimensional submodule for G i . By basicrepresentation theory, C G i ( S i ) is cyclic of order q −
1, and N G i ( S i ) /C G i ( S i ) is cyclicof order 4. Thus N G ( A ∗ ) = N G ( S ∗ ) ∼ = (( q − · × ( q − · h t, f i × GL (3 ) . Since A F = h f i acts as a field automorphisms, similar considerations show that N G ( A ) = S ( N G ( S ∗ ) ∩ C G ∗ ( f )) ∼ = (400 · × · h t, f i × GL (3) . The G -fusion in S is effected by the group S (4 × h t i , which is the same for bothnormalizers. In this example we may choose D = [ C G ( S ∗ ) , f ], which is of type B × B × ( SL ( q ) · B is cyclic of order ( q − / − D could be chosen inside the abelian factor B × B .4.1. Simple groups.
The following is an immediate consequence of Theorems 1.1 and 1.2 (here O A ( G ) =1 by the simplicity of G ): Corollary 4.1.
Let G be a simple group in which A ( S ) = S . Then G is isomorphicto one of the groups L i that appear in the conclusions of Theorems 1.1 and 1.2. Inall cases the normalizer of S controls strong fusion in S .Proof. The first assertion is immediate. Recall that if G is the simple group G ( q ) forsome q with ( q,
3) = 1, then we showed in the proof of Proposition 2.7 (and at theend of the proof of Lemma 3.4) that S = Ω ( S ). Thus by Corollary 1.4, in all casesin where A ( S ) = S the normalizer of S controls strong fusion in S . (cid:3) With the exception of the groups of Lie type in characteristic = p , the Sylow- p normalizers of the simple groups appearing in the conclusions to Theorems 1.1 and1.2 are described explicitly in Proposition 2.7. We therefore add here only someobservations on the structure of the normalizers in the remaining case.Let G be a group of Lie type over a field of characteristic r = p and supposethe Sylow p -subgroup S of G is abelian but not elementary abelian (here p is odd).The overall structure of N G ( S ) is governed by the theory of algebraic groups, asinvoked in the proof of Proposition 2.6. Recapping from that argument: since the TRONGLY CLOSED SUBGROUPS 37
Schur multiplier of G is prime to p we may work in the universal version of G todescribe N G ( S ). Let G be the simply connected universal simple algebraic groupover the algebraic closure of F r , and let σ be a Steinberg endomorphism whose fixedpoints equal G . In the notation of [SS70], p is not a torsion prime for G , so by 5.8therein C G ( S ) is a connected, reductive group whose semisimple component is simplyconnected. The general theory of connected, reductive algebraic groups gives that C G ( S ) = Z L , where Z is the connected component of the center of C G ( S ), L is thesemisimple component (possibly trivial), and Z ∩ L is a finite group. Furthermore, L is a product of groups of Lie type over the algebraic closure of F r of smaller rankthan G . It follows that C G ( S ) is a commuting product of the fixed points of σ on Z and L , i.e., C G ( S ) = C Z ( σ ) C L ( σ )where S ≤ C Z ( σ ) is an abelian group (a finite torus) and C L ( σ ) is either solvable ora product of finite Lie type groups in characteristic r .To complete the generic description of N G ( S ) we invoke additional facts from [SS70]and [GLSv3, Section 4.10]. As above, S is contained in a σ -stable maximal torus T ,where T is obtained from a σ -stable split maximal torus T by twisting by someelement w of the Weyl group W = N G ( T ) /T of G . Since S is characteristic in thefinite torus T = ( T ) σ it follows that N G ( S ) /C G ( S ) ∼ = N G ( T ) /T . In most cases, by1.8 of [SS70] or Proposition 3.36 of [Ca85] we have N G ( T ) /T ∼ = W σ ∼ = C W ( w ) (seealso [GLSv3, Theorem 2.1.2(d)] and the techniques in the proof of Theorem 4.10.2 inthat 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 underlyingnatural module, V , as described in the proof of Lemma 3.4. In the notation of thislemma, the semisimple component of order prime to p comes from the normal sub-group Isom( V ) in Isom( V ), where V = C V ( S ), and S is the direct product of thecyclic groups S ∩ Isom( V i ) for i = 1 , , . . . , s . The Weyl group normalizing S acts asthe symmetric group S s permuting the subgroups Isom( V i ). The orders of the central-izer and normalizer of a (cyclic) Sylow p -subgroup in each subgroup Isom( V i ) dependon p and the nature of G — Chapter 3 of [Ca85] gives techniques for computing these.For an easy explicit example of this let G = SL n +1 ( q ) where q = r m and p > n + 1,and assume p (cid:12)(cid:12) q −
1. In this case we may choose S contained in the group of diagonalmatrices T of determinant 1, which is an abelian group of type ( q − , . . . , q −
1) of rank n (here T is the split torus). In this case T = C G ( S ) and N G ( S ) = N G ( T ) = T W ,where W ∼ = S n +1 is the group of permutation matrices permuting the entries ofmatrices in T in the natural fashion (as the “trace zero” submodule of the naturalaction on the direct product of n + 1 copies of the cyclic group of order q − p -normalizer in the simple group P SL n +1 ( q ) factor out the subgroupof scalar matrices of order ( n + 1 , q − Split extensions.
In this subsection we consider some non-simple groups possessing strongly closed p -subgroups in which S = A ( S ). We show that many split extensions for which theseconditions hold can be constructed. This construction demonstrates that even when N G ( S ) (or N G ( A )) controls G -fusion in S , where overbars denote passage to G/O A ( G ),it need not be the case that N G ( A ) controls fusion in S (or in A ), even when N G ( A ) = N G ( A ). This highlights the importance of “recognizing” the subgroup O A ( G ) as wellas the isomorphism types of the components of G/O A ( G ) in our classifications. Proposition 4.2.
Let R be any group that is not a p -group but is generated byelements of order p . Assume also that A ( T ) = 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/C R ( E ) is not a p -group. Let G be the semidirect product E ⋊ R , and let S = ET be a Sylow p -subgroup of G . Then G is generated by elements of order p , A ( S ) = S ,and N G ( S ) does not control fusion in S .Proof. Note that the split extension G = ER is clearly generated by elements oforder p since both E and R are. Also, A ( S ) contains E , and by Lemma 2.3, sincethe extension is split we obtain A ( S ) /E ∼ = A ( T ) < T , so A ( S ) = S . It remains toshow that N G ( S ) does not control fusion in S .Let 0 = E < E < · · · < E n − < E n = E be a chief series through E , so that eachfactor E i /E i − is an irreducible F p R -module. If each such factor is one-dimensional,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 matrices,hence R/C R ( E ) is a p -group, a contradiction.Thus there is some chief factor E i /E i − that is not one-dimensional. If a Sylownormalizer controlled fusion in S , then by Lemma 2.3 the same would be true inthe quotient group G/E i − ; we show this is not the case. To do so, we may pass tothe quotient and therefore assume E is a minimal normal, noncentral subgroup of TRONGLY CLOSED SUBGROUPS 39 G . Now Z = Z ( S ) ∩ E = 1 and Z is invariant under N G ( S ). However, R actsirreducibly and nontrivially on E and R is generated by conjugates of S , so Z = E and hence Z is not R -invariant. Thus for some z ∈ Z and g ∈ G we must have z g ∈ E − Z , which shows N G ( S ) does not control fusion in S . (cid:3) This proposition can be invoked to create a host of examples: Let R be any ofthe simple groups L i (or their quasisimple universal covers) in the conclusion toTheorem 1.2 and let E be an F p R -module on which R acts nontrivially (for example,any nontrivial permutation module). More specifically, for p odd let q be any primepower such that p (cid:12)(cid:12) q −
1, so that Sylow p -subgroups of R = SL ( q ) are cyclicof order ≥ p (for example, p = 3 and q = 19). Then R permutes the q + 1 linesin a 2-dimensional space over F q , and so permutes q + 1 basis vectors in a q + 1-dimensional vector space E over F p . Then G = E ⋊ R gives a specific realization forProposition 4.2.Building on the preceding example where R = SL (2 , q ) for any prime power q such that p (cid:12)(cid:12) q −
1: then T may be represented by diagonal matrices over F q , so iscyclic of order p n = | q − | p ; moreover, C R ( T ) is the group of all diagonal matrices ofdeterminant 1, hence is cyclic of order q −
1. In particular, A ( T ) = Ω ( T ) ∼ = Z /p .Furthermore, N R ( T ) = N R ( A ( T )) is of index 2 in C R ( T ) and an involution in N R ( T )inverts C R ( T ). Thus N R ( A ( T )) / A ( T ) is isomorphic to the dihedral group of order2( q − /p .4.3. Exotic extensions of G ( q ) G ( q ) G ( q ) . When G is the simple group G ( q ) for some q with ( q,
3) = 1, although a Sylow3-subgroup S contains a strongly closed subgroup A = Z ( S ) of order p = 3, when weimpose the additional hypothesis that our strongly closed subgroup must contain allelements of order 3 the strongly closed subgroup A does not arise in our considerationsbecause S = Ω ( S ). For the same reason, if G = ER is any split extension of R = G ( q ) by an elementary abelian 3-group and S = ET for T ∈ Syl ( R ), thenagain S = Ω ( S ) = A ( S ). In this subsection we describe a family of extensionsthat we call “half-split” in the sense that they split over a certain conjugacy class ofelements of R but do not split over another. In this way we construct extensions G of R = G ( q ) by certain elementary abelian 3-groups E such that for S ∈ Syl ( G )we have Ω ( S ) /E mapping onto the strongly closed subgroup of order 3 in a Sylow3-subgroup S/E of G ( q ). In particular, these “exotic” extensions show that the exceptional case of Corollary 1.4 cannot be removed: when 9 (cid:12)(cid:12) q − G are generated by elements of order 3, have A ( S ) = S , but N G/E ( S/E ) does notcontrol fusion in
S/E (here E = O A ( G ) where A = A ( S )).The following general proposition will construct such extensions. Proposition 4.3.
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 F p R -module E and an extension −→ E −→ G −→ R −→ 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 . Inparticular, for nonidentity elements x ∈ X and z ∈ Z every element in the coset xE has order p whereas zE contains elements of order p in G .Proof. Let E be the one-dimensional trivial F p X -module. By the familiar cohomol-ogy of cyclic groups ([Bro82], Section III.1):(4.1) H ( X, E ) ∼ = Z /p Z and a non-split extension of X by E is just a cyclic group of order p . Now let E = Coind RX E = Hom Z X ( Z R, E )be the coinduced module from X to R (which is isomorphic to the induced module E ⊗ F p X F p R in the case of finite groups), so that E has F p -dimension p | R | . ByShapiro’s Lemma ([Bro82], Proposition III.6.2)(4.2) H ( R, E ) ∼ = H ( X, E ) . Thus by (4.1) there is a non-split extension of R by E — call this extension group G and identify E as a normal subgroup of G with quotient group G/E = R .The isomorphism in Shapiro’s Lemma, (4.2), is given by the compatible homomor-phisms ι : X ֒ → R and π : Coind RX E → E , where π is the natural map π ( f ) = f (1).In particular, this isomorphism is a composition H ( R, E ) res −→ H ( X, E ) π ∗ −→ H ( X, E ) . Thus the 2-cocycle defining the non-split extension group G , which maps to a non-trivial element in H ( X, E ), by restriction gives a non-split extension of X by E aswell. TRONGLY CLOSED SUBGROUPS 41
For any subgroup Z of R of order p with Z not conjugate to X , by the Mackeydecomposition for induced representations(4.3) Res RZ Ind RX E = M g ∈R Ind ZZ ∩ gXg − Res gXg − Z ∩ gXg − gE where R is a set of representatives for the ( Z, X )-double cosets in R . By hypothesis, Z ∩ gXg − = 1 for every g ∈ R , hence each term in the direct sum on the right handside is an F p Z -module obtained by inducing a one-dimensional trivial F p -module forthe identity subgroup to a p -dimensional F p Z -module, i.e., is a free F p Z -module ofrank 1. (Alternatively, E is the F p -permutation module for the action of R by leftmultiplication on the left cosets of X ; by the fusion hypothesis, Z acts on a basis of E as a product of disjoint p -cycles with no 1-cycles.) This shows E is a free F p Z -module,and hence the extension of Z by E splits. This completes the proof. (cid:3) The p th -power map on elements in the lift of X to G can be described more precisely.By the Mackey decomposition in (4.3) inducing from X but rather restricting to X instead of Z , or by direct inspection of the action of X on the F p -permutation module E , we see that E decomposes as an F p X -module direct sum as E = E ⊕ E , where E is a trivial F p X -module and E is a free F p X -module. Since X splits overthe free summand E , we see that X does not split over E , and hence XE ∼ = ( Z /p ) × Z /p × · · · × Z /p with E = Ω ( XE ) . Thus for every element x in G − E mapping to an element of X in G/E , x p has anontrivial component in E .One may also observe that by taking direct sums we can arrange more generally thatif X , X , . . . , X n are representatives of the distinct conjugacy classes of subgroups oforder p in R , then for any i ∈ { , , . . . , n } there is an F p R -module E and an extensionof R by E such that in the extension group each of X , . . . , X i splits over E but noneof X i +1 , . . . , X n do.We are particularly interested in the case R = G ( q ) with p = 3 and ( q,
3) = 1.The normalizer of a Sylow 3-subgroup of R is described in Proposition 2.7: Let T ∈ Syl ( R ) and let Z = Z ( T ) = h z i . In the notation preceding Proposition 2.5, N R ( Z ) ∼ = SL ǫ ( q ) · (cid:12)(cid:12) q − ǫ . Moreover, if 9 (cid:12)(cid:12) q − ǫ then N R ( T ) does not control fusion in T : all elements of order 3 in T − Z are conjugate in C R ( Z ) whereasby Proposition 2.7, N R ( T ) /T has order 4 for this congruence of q .Now consider the extension group G constructed in Proposition 4.3 with p = 3, R = G ( q ), Z = h z i and X = h x i for any x ∈ T − Z of order 3. Let S ∈ Syl ( G ) with S mapping onto T in G/E ∼ = R . Since Proposition 2.7 shows all elements of order3 in T − Z are conjugate to x but not to z , the structure of the extension impliesthat A = Ω ( S ) = A ( S ) contains E and maps to Z in S/E . Thus O A ( G ) = E and A = Z . By Corollary 1.4, the normalizer of Z in R = G ( q ) controls 3-fusionin G ( q ), so in particular SL ∗ ( q ) has the same mod 3 cohomology as G ( q ), where SL ∗ ( q ) denotes the group SL ǫ ( q ) together with the outer (graph) automorphism oforder 2 inverting its center ( N R ( Z ) ∼ = SL ∗ ( q )). On the other hand, Z is normal in SL ∗ ( q ), and SL ∗ ( q ) /Z is isomorphic to P SL ∗ ( q ).This example highlights the importance of having a classification of all groupspossessing a nontrivial strongly closed p -subgroup that is not Sylow — not just thesimple groups having such a subgroup that contains Ω ( S ) — since the subgroup A ( S ) does not pass in a transparent fashion to quotients.The extensions of our techniques and results to more general p -local spaces witha notion of p -fusion seem to be the natural next step of our study; in particular,classifying spaces of p -local finite groups and some families of non-finite groups offerenticing possibilities. References [Asc94] M. Aschbacher.
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Ram´on J. FloresDepartamento de Estad´ıstica, Universidad Carlos III de Madrid, C/ Madrid 126E – 28903 Colmenarejo (Madrid) — Spaine-mail: [email protected]