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