A geometric approach to Quillen’s conjecture

Antonio Díaz Ramos; Nadia Mazza

doi:10.1515/jgth-2021-0033

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A geometric approach to Quillen’s conjecture

Antonio Díaz Ramos and Nadia Mazza

Published/Copyright: September 16, 2021

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From the journal Journal of Group Theory Volume 25 Issue 1

Abstract

We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of | G | . Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex, Ann. of Math. (2) 137 (1993), 3, 473–529], our techniques are simpler.

1 Introduction

Let 𝐺 be a finite group, and let 𝑝 be a prime number dividing the order of 𝐺. Let A p ⁢ ( G ) be the poset of all non-trivial elementary abelian 𝑝-subgroups of 𝐺 ordered by inclusion, and let | A p ⁢ ( G ) | be its realization as a topological space. This space is a simplicial complex which has as 𝑛-simplices the chains of length 𝑛 in A p ⁢ ( G ) :

P 0 < P 1 < ⋯ < P n , with ⁢ P i ∈ A p ⁢ ( G ) .

We denote by O p ⁢ ( G ) the largest normal 𝑝-subgroup of 𝐺, and by rk p ⁡ ( G ) the 𝑝-rank of 𝐺, that is, the maximum of the ranks of the elementary abelian 𝑝-subgroups of 𝐺. A finite elementary abelian 𝑝-group 𝐸 has rank 𝑟 if | E | = p r , or equivalently, r = dim F p ⁡ E when we regard 𝐸 as an F p -vector space.

Conjecture 1.1

Conjecture 1.1 (Quillen’s conjecture [10])

If | A p ⁢ ( G ) | is contractible, then we have O p ⁢ ( G ) ≠ 1 .

Quillen stated the conjecture in his seminal paper [10], in which he also proved the conjecture for finite groups of 𝑝-rank at most 2 [10, Proposition 2.10], for finite groups of Lie type in defining characteristic [10, Theorem 3.1] and for solvable finite groups [10, Theorem 12.1]. Recently, Piterman, Sadofschi Costa and Viruel [9] proved Quillen’s conjecture for finite groups of 𝑝-rank at most 3. Quillen’s results were extended to 𝑝-solvable finite groups by several people using different methods, notably by Alperin, using results on coprime actions and the Classification of the Finite Simple Groups [12, Theorem 8.2.12], and by Hawkes and Isaacs in the particular case of 𝑝-solvable finite groups with abelian Sylow 𝑝-subgroups, via a combinatorial method and without resorting to the Classification [6, Theorem A]. In [2], Aschbacher and Smith tackled Quillen’s conjecture via reduction theorems based on the assumption of the existence of a counter-example to the conjecture of minimal order. Thus, they showed that Quillen’s conjecture holds for 𝑝-solvable finite groups in another way, and their overarching results are the more general ones to date: if p > 5 and 𝐺 does not have a unitary component U n ⁢ ( q ) with q ≡ - 1 ( mod p ) and 𝑞 odd, then 𝐺 satisfies Quillen’s conjecture [2, Main Theorem, p. 474]. In their work, the authors introduced the following property involving reduced homology with rational coefficients. Here, we employ integral coefficients.

Definition 1.2

Definition 1.2 ( Q ⁢ D p )

The finite group 𝐺 with r = rk p ⁡ ( G ) has the Quillen dimension at 𝑝 property, written Q ⁢ D p , if

H ~ r - 1 ⁢ ( | A p ⁢ ( G ) | ) ≠ 0 .

By H ~ * ⁢ ( | A p ⁢ ( G ) | ) , we mean reduced integral simplicial homology [11, Chapter 7], i.e.,

H ~ * ⁢ ( | A p ⁢ ( G ) | ) = H * ( | A p ( G ) | ; Z ) for * > 0 , H 0 ⁢ ( | A p ⁢ ( G ) | ; Z ) = H ~ 0 ⁢ ( | A p ⁢ ( G ) | ) ⊕ Z .

Note that, as | A p ⁢ ( G ) | is an ( r - 1 ) -dimensional complex, r - 1 is the top dimension for which reduced homology can possibly be non-zero, and that, in this dimension, the reduced homology group is necessarily a free abelian group. Hence, the rational and integral versions of this property are equivalent. In addition, by [8, Theorem 2], Quillen’s conjecture is equivalent to the acyclicity condition that H ~ * ⁢ ( | A p ⁢ ( G ) | ) ≠ 0 .

In [2, Theorem 3.1], they consider 𝑝-extensions of finite simple groups, that is, almost-simple groups with an elementary abelian 𝑝-group inducing outer automorphisms. Aschbacher and Smith prove that most of these 𝑝-extensions satisfy Q ⁢ D p , and they list those which do not. In [3], the first author proves that Quillen’s conjecture holds for solvable and 𝑝-solvable finite groups via new geometrical methods. In the present paper, we elaborate on these methods with the objective of finding shorter and easier proofs of the results in [2]. We deal with the alternating, symmetric and classical finite groups in non-defining characteristic and for an odd prime 𝑝. By a finite classical group, we mean a finite linear, unitary, symplectic or orthogonal group. With these methods, we prove the following result.

Theorem 1.3

Let 𝑝 be an odd prime, and let 𝐺 be one of the following groups:

𝐺 is an alternating or symmetric group of degree 𝑛 for p ≥ 5 , and for p = 3 , n = 4 , 5 , 8 .
𝐺 is a linear, unitary, symplectic or orthogonal group defined over a field of characteristic different from 𝑝, unless p ∣ ( q + 1 ) and 𝐺 is a unitary group defined over F q 2 or 𝐺 is a symplectic or orthogonal group defined over F 2 or G = PSL 3 ⁡ ( q ) with q ≡ 1 ( mod 3 ) .

Then 𝐺 has the Quillen dimension at 𝑝 property, and therefore Quillen’s conjecture holds for 𝐺.

The structure of the paper is as follows. In Section 2, we define faithful collections and discuss some of their properties. Such a collection for a group 𝐺 consists of an arrangement of elementary abelian 𝑝-subgroups of 𝐺 together with certain elements of 𝐺 that centralize/normalize these 𝑝-subgroups. In Section 3, we find further conditions on faithful collections which imply that a finite group has the Quillen dimension at 𝑝 property Q ⁢ D p , and we call such faithful collections admissible. In Section 4, we briefly study when the Quillen dimension at 𝑝 property of a given finite group is inherited by its quotient groups. Then we show the Quillen dimension at 𝑝 property for the symmetric and alternating groups, and for the finite classical groups in non-defining characteristic, excluding certain cases when p = 3 . Thus Theorem 1.3 summarizes these results. We also present some limitations of our method and open questions.

2 Faithful collections

In this section, we define faithful collections for a finite group 𝐺.

Given E ∈ A p ⁢ ( G ) , we regard 𝐸 as an F p -vector space (generally written additively), and a faithful collection for 𝐺, if it exists, is a certain arrangement of hyperplanes of 𝐸 and elements of 𝐺 subject to certain constraints.

Definition 2.1

Let 𝑟 be a positive integer. For any integer 𝑙 with 0 ≤ l ≤ r , we define an 𝑙-tuple for 𝑟 to be an ordered sequence of integers i = [ i 1 , … , i l ] with 1 ≤ i j ≤ r and no repetition. By S l r , we denote the set of all 𝑙-tuples for 𝑟 for a given 𝑙, and by S r = ⋃ l = 0 , … , r S l r the set of all 𝑙-tuples for 𝑟 for all 0 ≤ l ≤ r .

For l = r , the unique 𝑟-tuple is [ 1 , 2 , … , r ] , and for l = 0 , the unique 0-tuple corresponds to the empty sequence, and we denote it by [ ] . Thus,

S r r = { [ 1 , 2 , … , r ] } and S 0 r = { [ ] } .

Definition 2.2

Let E = ⟨ e 1 , … , e r ⟩ be an elementary abelian 𝑝-subgroup of 𝐺 of rank r ≤ rk p ⁡ ( G ) . For each 𝑙-tuple i = [ i 1 , … , i l ] ∈ S l r , set

E i = E [ i 1 , … , i l ] = ⟨ e 1 , … , e i 1 ^ , … , e i l ^ , … , e r ⟩

for the subgroup of 𝐸 generated by all the e i , except e i 1 , … , e i l .

We abuse notation and write E i = E [ i ] = ⟨ e 1 , … , e i ^ , … , e r ⟩ . In Definition 2.2, the subgroups E i are subspaces of 𝐸 of codimension 1. Note that we have the following:

E [ i 1 , … , i l ] = E i 1 ∩ ⋯ ∩ E i l , E [ ] = E = ⟨ E i , e i ⟩ , E [ 1 , 2 , … , r ] = { 0 } ,

and in particular,

(2.1) ⟨ e i ⟩ = E [ 1 , … , i ^ , … , r ] = E 1 ∩ ⋯ ∩ E i ^ ∩ ⋯ ∩ E r .

Remark 2.3

Note that

E [ i 1 , … , i l ] = E [ i 1 ′ , … , i l ′ ′ ] ⇔ l = l ′ and { i 1 , … , i l } = { i 1 ′ , … , i l ′ ′ } .

Definition 2.4

Definition 2.4 (Faithful collection)

Let E = ⟨ e 1 , … , e r ⟩ be an elementary abelian 𝑝-subgroup of 𝐺 of rank r ≤ rk p ⁡ ( G ) , and let c = ( c 1 , … , c r ) be an ordered 𝑟-tuple of elements of 𝐺. Suppose that, for any i ∈ S r and for any ordered 𝑟-tuple ϵ = ( ϵ 1 , … , ϵ r ) of integers ϵ i ∈ { - 1 , 0 , 1 } , we have

E i c ϵ ≤ E ⟹ E i c ϵ = E i , where ⁢ E i c ϵ = E i c 1 ϵ 1 ⁢ … ⁢ c r ϵ r .

Then we say that

{ E i , c j } ⁢ is a faithful collection .

Note that the subgroups E i = E [ i ] = ⟨ e 1 , … , e i ^ , … , e r ⟩ of 𝐸 in a given faithful collection determine the subgroups ⟨ e i ⟩ by equation (2.1), and hence also the subgroups E i for any i ∈ S r . So { E i , c j } is a faithful collection if the subspaces E i of 𝐸 in Definition 2.2 can be conjugated by the elements c ϵ of 𝐺 to a subspace of 𝐸 if and only if c ϵ normalizes E i . This mimics a property of coprime actions; see [3, Proposition 2.2 (1)]. Note that if i ∈ S 0 r or S r r , the implication

E i c ϵ ≤ E ⟹ E i c ϵ = E i .

is vacuous.

The next result is a characterization of faithful collections in terms of the elements e i generating 𝐸 in Definition 2.2. By equation (2.1), these elements are determined by the hyperplanes E i up to a power.

Lemma 2.5

Let E = ⟨ e 1 , … , e r ⟩ be an elementary abelian 𝑝-subgroup of 𝐺 of rank 𝑟. Let c = ( c 1 , … , c r ) ∈ G r . Then { E i , c j } is a faithful collection if and only if, for any ϵ ∈ { - 1 , 0 , 1 } r and any 1 ≤ i ≤ r , we have

⟨ e i ⟩ c ϵ ≤ E ⟹ ⟨ e i ⟩ c ϵ = ⟨ e i ⟩ .

Proof

That a faithful collection satisfies the given conditions is clear by considering the ( r - 1 ) -tuples [ 1 , … , i ^ , … , r ] . Conversely, let i ∈ S r - l r for some 0 ≤ l ≤ r , and let ϵ = ( ϵ 1 , … , ϵ r ) ∈ { - 1 , 0 , 1 } r such that E i c ϵ ≤ E , where E i = ⟨ e j 1 , … , e j l ⟩ has rank 𝑙. In particular, we have

⟨ e j k ⟩ c ϵ ≤ E

for 1 ≤ k ≤ l , and so, by hypothesis, ⟨ e j k ⟩ c ϵ = ⟨ e j k ⟩ for 1 ≤ k ≤ l . It follows that E i c ϵ = ⟨ e j 1 , … , e j l ⟩ c ϵ = ⟨ e j 1 c ϵ , … , e j l c ϵ ⟩ = ⟨ e j 1 , … , e j l ⟩ = E i . ∎

The next result gives sufficient conditions for the existence of a faithful collection.

Lemma 2.6

Let E = ⟨ e 1 , … , e r ⟩ be an elementary abelian 𝑝-subgroup of 𝐺 of rank 𝑟, and let c = ( c 1 , … , c r ) ∈ G r . Suppose that c i ∈ C G ⁢ ( E i ) ∖ C G ⁢ ( e i ) and [ c i , c j ] = 1 for all i , j . Then { E i , c j } is a faithful collection.

Proof

We use Lemma 2.5. Consider ϵ = ( ϵ 1 , … , ϵ r ) with ϵ i ∈ { - 1 , 0 , 1 } , and let 1 ≤ i 0 ≤ r such that ⟨ e i 0 ⟩ c ϵ ≤ E . Then we have

e i 0 c ϵ = e i 0 c 1 ϵ 1 ⁢ … ⁢ c r ϵ r = e i 0 c i 0 ϵ i 0 = λ 1 ⁢ e 1 + ⋯ + λ i 0 ⁢ e i 0 + ⋯ + λ r ⁢ e r ,

where λ i ∈ F p (not all zero), and in the second equality, we have used the facts that c i ∈ C G ⁢ ( E i ) , that [ c i , c i 0 ] = 1 and that e i 0 ∈ E i for i ≠ i 0 . Assume λ i 1 ≠ 0 for some i 1 ≠ i 0 . Then

E = ⟨ E i 1 , e i 0 c i 0 ϵ i 0 ⟩ .

Now, by hypothesis, c i 1 centralizes e i 0 ∈ E i 1 , and we calculate

e i 0 c i 1 ⁢ c i 0 ϵ i 0 = e i 0 c i 0 ϵ i 0 ⁢ c i 1 = e i 0 c i 0 ϵ i 0 ,

using the facts that [ c i 1 , c i 0 ] = 1 , that c i 1 ∈ C G ⁢ ( E i 1 ) and that e i 0 ∈ E i 1 (as i 0 ≠ i 1 ). This is a contradiction with c i 1 ∉ C G ⁢ ( E ) , and hence λ i = 0 for all i ≠ i 0 and e i 0 c ϵ = λ i 0 ⁢ e i 0 ∈ ⟨ e i 0 ⟩ . ∎

In the next section, we will use the existence of faithful collections with 𝐸 of rank r = rk p ⁡ ( G ) subject to certain constraints to prove that Q ⁢ D p holds for 𝐺; see Theorem 3.5.

By contrast, we prove that, under certain assumptions on 𝐺, there cannot be a faithful collection subject to the conditions in Lemma 2.6. Recall that a maximal elementary abelian 𝑝-subgroup of 𝐺 is a maximal element in the poset A p ⁢ ( G ) , i.e., an elementary abelian 𝑝-subgroup 𝐸 of 𝐺 which is not properly contained in any other elementary abelian 𝑝-subgroup of 𝐺. But 𝐸 need not have maximal rank rk p ⁡ ( G ) .

Lemma 2.7

Suppose that O p ⁢ ( G ) ∩ Z ⁢ ( G ) > 1 for some odd prime 𝑝. Let 𝐸 be a maximal elementary abelian 𝑝-subgroup of 𝐺, not necessarily of maximal order. Then there exists no collection { E i , c j } for 𝐸 subject to c i ∈ C G ⁢ ( E i ) ∖ C G ⁢ ( e i ) for all 𝑖.

Proof

Let E = ⟨ e 1 , … , e r ⟩ ∈ A p ⁢ ( P ) be a maximal element, and suppose that { E i , c j } is a collection subject to c i ∈ C G ⁢ ( E i ) ∖ C G ⁢ ( e i ) for all 𝑖. Let

V = Ω 1 ⁢ ( O p ⁢ ( G ) ∩ Z ⁢ ( G ) ) .

Note that E = V ⁢ E ≥ V by maximality of 𝐸 and the fact that 𝑉 lies in the center of every Sylow 𝑝-subgroup of 𝐺.

Let 1 ≠ v = ∑ i = 1 r λ i ⁢ e i ∈ V , and without loss, suppose λ 1 = 1 . (Here we use the additive notation of 𝐸 seen as vector space.) So v = e 1 + v ′ with v ′ ∈ E 1 . Because v ∈ V ≤ Z ⁢ ( G ) and c 1 ∈ C G ⁢ ( E 1 ) , we have

v = v c 1 = ( e 1 + v ′ ) c 1 = ( e 1 c 1 ) + v ′ .

Therefore, e 1 = v - v ′ = e 1 c 1 , showing that c 1 ∈ C G ⁢ ( e 1 ) , a contradiction. ∎

3 Generalization

We want to find sufficient conditions which imply that 𝐺 has the Quillen dimension at 𝑝 property, or more generally, which imply that Quillen’s conjecture holds for 𝐺. Theorem 3.5 below is a generalization of [3, Theorem 5.3].

Definition 3.1

For an 𝑙-tuple i = [ i 1 , … , i l ] ∈ S l r , we define the signature

sgn ⁡ ( i ) = ( - 1 ) n + m ⁢ of ⁢ i ,

where

𝑛 is the number of transpositions we need to apply to the 𝑙-tuple 𝐢 to rearrange it in increasing order [ j 1 , … , j l ] , and
𝑚 is the number of positions in which [ j 1 , … , j l ] differ from [ 1 , … , l ] .

Note that, in Definition 3.1, the number 𝑛 of transpositions is not uniquely defined, but its parity is.

For instance, if i = [ 1 , 4 , 2 ] , then n = 1 since we need to apply ( 2 , 4 ) to reorder 𝐢 as [ 1 , 2 , 4 ] , and m = 1 since [ 1 , 2 , 4 ] differs from [ 1 , 2 , 3 ] only in one place. Thus sgn ⁡ ( i ) = 1 .

Let E = ⟨ e 1 , … , e r ⟩ be an elementary abelian 𝑝-subgroup of rank 𝑟 of the group 𝐺. We now generalize the chains introduced in [3, Section 3] to the case when 𝐸 need not have maximal rank rk p ⁡ ( G ) and 𝐺 need not be a semi-direct product. We consider the poset A p ⁢ ( E ) and its order complex Δ ⁢ ( A p ⁢ ( E ) ) . We define an element of the integral simplicial chains of dimension r - 1 , C r - 1 ⁢ ( Δ ⁢ ( A p ⁢ ( E ) ) ) . For i = [ i 1 , … , i r - 1 ] ∈ S r - 1 r , we define the ( r - 1 ) -simplex

σ i = ( E [ i 1 , … , i r - 1 ] < E [ i 1 , … , i r - 2 ] < ⋯ < E i 1 < E ) ∈ Δ ( A p ( E ) ) ,

and for a ∈ Z , the chain

Z E , a = a ⁢ ∑ i ∈ S r - 1 r sgn ⁡ ( i ) ⁢ σ i in ⁢ C r - 1 ⁢ ( Δ ⁢ ( A p ⁢ ( E ) ) ) .

By definition of the differential,

d ⁢ ( Z E , a ) = a ⁢ ∑ 0 ≤ j < r ∑ i ∈ S r - 1 r ( - 1 ) j ⁢ sgn ⁡ ( i ) ⁢ d j ⁢ ( σ i ) ,

where

d j ( σ i ) = ( E [ i 1 , … , i r - 1 ] < ⋯ < E [ i 1 , … , i r - j ] < E [ i 1 , … , i r - j - 2 ] < ⋯ < E i 1 < E )

removes the ( j + 1 ) -th term from the left in the chain for 0 ≤ j < r . So, in particular,

d 0 ( σ i ) = ( E [ i 1 , … , i r - 2 ] < ⋯ < E ) and d r - 1 ( σ i ) = ( E [ i 1 , … , i r - 1 ] < ⋯ < E i 1 ) .

Hence, define the ( r - 2 ) -simplex

τ i = d r - 1 ( σ i ) = ( E [ i 1 , … , i r - 1 ] < ⋯ < E i 1 ) .

Let us recall the following useful property [3, Proposition 3.2].

Proposition 3.2

With the above notation,

d ⁢ ( Z E , a ) = ( - 1 ) r - 1 ⁢ a ⁢ ∑ i ∈ S r - 1 r sgn ⁡ ( i ) ⁢ τ i .

Proof

Suppose that d k ⁢ ( σ i ) = d l ⁢ ( σ j ) for some i , j ∈ S r - 1 r and 0 ≤ k , l ≤ r - 1 , that is,

( E [ i 1 , … , i r - 1 ] < ⋯ < E [ i 1 , … , i r - k ] < E [ i 1 , … , i r - k - 2 ] < ⋯ < E i 1 < E ) = ( E [ j 1 , … , j r - 1 ] < ⋯ < E [ j 1 , … , i r - l ] < E [ j 1 , … , j r - l - 2 ] < ⋯ < E j 1 < E ) .

For both chains to be equal, the jump by an index p 2 must occur in the same place, saying that k = l . If 0 < k = l < r - 1 , then, by Remark 2.3, the tuples 𝐢 and 𝐣 are identical but for { i r - k - 1 , i r - k } = { j r - k - 1 , j r - k } . So either i = j , or 𝐢 and 𝐣 differ by one transposition, and hence sgn ⁡ ( i ) = - sgn ⁡ ( j ) . In the latter case, the corresponding summands sgn ⁡ ( i ) ⁢ d k ⁢ ( σ i ) and sgn ⁡ ( j ) ⁢ d l ⁢ ( σ j ) add up to zero. Assume now that k = l = 0 . Then, again by Remark 2.3, [ j 1 , … , j r - 2 ] = [ i 1 , … , i r - 2 ] and either i = j or j r - 1 ≠ i r - 1 . In the latter case, by [3, Lemma 2.4], sgn ⁡ ( i ) = - sgn ⁡ ( j ) , and again, the two terms cancel each other out. Finally, if k = l = r - 1 , the tuples 𝐢 and 𝐣 are identical and the terms contribute to the sum in the statement of the proposition. ∎

Consider the order complex Δ ⁢ ( A p ⁢ ( G ) ) . Then the group 𝐺 acts by conjugation on C * ⁢ ( Δ ⁢ ( A p ⁢ ( G ) ) ) . For x ∈ G , the element

Z E , a x = a ∑ i ∈ S r - 1 r sgn ( i ) σ i x = a ∑ i ∈ S r - 1 r sgn ( i ) ( E [ i 1 , … , i r - 1 ] x < ⋯ < E i 1 x < E x )

belongs to Δ ⁢ ( A p ⁢ ( E x ) ) ⊆ Δ ⁢ ( A p ⁢ ( G ) ) .

Let 𝐽 be a non-empty finite indexing set, and consider subsets x = { x j } j ∈ J ⊆ G and a = { a j } j ∈ J ⊆ Z . Define the chain in C r - 1 ⁢ ( Δ ⁢ ( A p ⁢ ( G ) ) ) ,

(3.1) Z G , x , a = ∑ j ∈ J Z E , a j x j = ∑ j ∈ J a j ⁢ ∑ i ∈ S r - 1 r sgn ⁡ ( i ) ⁢ σ i x j .

Then, by Proposition 3.2,

d ⁢ ( Z G , x , a ) = ( - 1 ) r - 1 ⁢ ∑ j ∈ J a j ⁢ ∑ i ∈ S r - 1 r sgn ⁡ ( i ) ⁢ τ i x j .

So, given j ∈ J and i ∈ S r - 1 r , the coefficients of σ i x j and τ i x j are respectively

(3.2) C j , i := ∑ ( l , k ) ∈ C ⁢ ( j , i ) a l ⁢ sgn ⁡ ( k ) ,

(3.3) D j , i := ( - 1 ) r - 1 ⁢ ∑ ( l , k ) ∈ D ⁢ ( j , i ) a l ⁢ sgn ⁡ ( k ) ,

where

C ⁢ ( j , i ) = { ( l , k ) ∈ J × S r - 1 r ∣ E [ k 1 , … , k t ] x l = E [ i 1 , … , i t ] x j ⁢ for all ⁢ 0 ≤ t < r } , D ⁢ ( j , i ) = { ( l , k ) ∈ J × S r - 1 r ∣ E [ k 1 , … , k t ] x l = E [ i 1 , … , i t ] x j ⁢ for all ⁢ 1 ≤ t < r } .

Note that C ⁢ ( j , i ) ⊆ D ⁢ ( j , i ) (and recall that E [ ] = E ).

If we further assume that 𝐸 is a maximal elementary abelian 𝑝-subgroup of 𝐺, we want to find sufficient conditions for the existence of a non-zero cycle in H ~ r - 1 ⁢ ( | A p ⁢ ( G ) | ) .

Theorem 3.3

Let E = ⟨ e 1 , … , e r ⟩ be a maximal elementary abelian 𝑝-subgroup of rank 𝑟 of the group 𝐺. If the subsets x = { x j } j ∈ J ⊆ G and a = { a j } j ∈ J ⊆ Z satisfy that C j , i ≠ 0 for some j ∈ J and some i ∈ S r - 1 r and that D j , i = 0 for all j ∈ J and all i ∈ S r - 1 r , then

0 ≠ [ Z G , x , a ] ∈ H ~ r - 1 ⁢ ( | A p ⁢ ( G ) | ) .

In particular, | A p ⁢ ( G ) | is not contractible and Quillen’s conjecture holds for 𝐺. If furthermore r = rk p ⁡ ( G ) , then Q ⁢ D p holds for 𝐺.

Proof

Consider the chain Z G , x , a ∈ C r - 1 ⁢ ( Δ ⁢ ( A p ⁢ ( G ) ) ) defined in (3.1). The condition in the statement for the coefficients C j , i is clearly equivalent to Z G , x , a ≠ 0 . The condition on the coefficients D j , i is equivalent to d ⁢ ( Z G , x , a ) = 0 (cf. also [3, Proposition 4.2]). By the maximality of 𝐸, this cycle cannot be a boundary, and hence it gives rise to a non-zero homology class in H ~ r - 1 ⁢ ( | A p ⁢ ( G ) | ) . ∎

The assumptions of Theorem 3.3 are fulfilled when we can find a faithful collection subject to certain constraints, as described in the next theorem.

Theorem 3.4

Let E = ⟨ e 1 , … , e r ⟩ be a maximal elementary abelian 𝑝-subgroup of 𝐺 of rank 𝑟, and let { E i , c j } be a faithful collection such that [ c i , c j ] = 1 for all i , j . Set J = { δ = ( δ 1 , … , δ r ) , δ i ∈ { 0 , 1 } } , let a = { a δ } δ ∈ J ⊆ Z , and consider the following subset of 𝐺:

x = { c δ = c 1 δ 1 ⁢ ⋯ ⁢ c r δ r ∣ δ ∈ J } .

Then, for each such δ ∈ J and each i = [ i 1 , … , i r - 1 ] ∈ S r - 1 r ,

we have
C δ , i = sgn ⁡ ( i ) ⁢ ( ∑ δ ′ a δ ′ ) ,
where the sum is over all δ ′ ∈ J such that c δ - δ ′ ∈ N G ⁢ ( E ) ;
we have
D δ , i = ( - 1 ) r - 1 ⁢ sgn ⁡ ( i ) ⁢ ( ∑ δ ′ a δ ′ ) ,
where the sum is over all δ ′ ∈ J such that c δ - δ ′ ∈ N G ⁢ ( E i 1 ) ;
if c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) for all 1 ≤ i ≤ r , then C δ , i = sgn ⁡ ( i ) ⁢ a δ ;
if c i ∈ N G ⁢ ( E i ) for all 1 ≤ i ≤ r , then
D δ , i = ( - 1 ) r - 1 ⁢ sgn ⁡ ( i ) ⁢ ( ∑ ( a δ ′ + a δ ′′ ) ) ,
where the sum is over the pairs δ ′ , δ ′′ ∈ J such that c δ - δ ′ , c δ - δ ′′ ∈ N G ⁢ ( E i 1 ) , δ j ′′ = δ j ′ for all j ≠ i 1 and δ i 1 ′ + δ i 1 ′′ = 1 .

Proof

With the faithfulness condition in Definition 2.4 and Remark 2.3, a straightforward computation shows that ( δ ′ , k ) ∈ C ⁢ ( δ , i ) (cf. the definition of C ⁢ ( δ , i ) in equation (3.2) above) if and only if k = i and c δ - δ ′ ∈ N G ⁢ ( E ) , and similarly, ( δ ′ , k ) ∈ D ⁢ ( δ , i ) (cf. the definition of D ⁢ ( δ , i ) in equation (3.3) above) if and only if k = i and c δ - δ ′ ∈ N G ⁢ ( E i 1 ) . In other words,

C ⁢ ( δ , i ) = { ( δ ′ , i ) ∈ J × S r - 1 r ∣ c δ - δ ′ ∈ N G ⁢ ( E ) } , D ⁢ ( δ , i ) = { ( δ ′ , i ) ∈ J × S r - 1 r ∣ c δ - δ ′ ∈ N G ⁢ ( E i 1 ) } .

From here, we immediately get assertions 1 and 2.

Suppose the hypotheses in point 3 hold. Again, since

E c δ = E c δ ′ ⇔ E c δ - δ ′ = E ,

where each δ j - δ j ′ ∈ { - 1 , 0 , + 1 } , the faithfulness condition implies that

⟨ e i ⟩ c δ - δ ′ = ⟨ e i ⟩ c i δ i - δ i ′ = ⟨ e i ⟩ ,

where we have used the facts that [ c i , c j ] = 1 , that c j ∈ C G ⁢ ( E j ) and that e i ∈ E j for i ≠ j . By assumption, c i ∉ N G ⁢ ( ⟨ e i ⟩ ) , which forces δ i = δ i ′ , and this holds for all 1 ≤ i ≤ r .

Finally, for point 4, let δ ′ be such that E i 1 c δ = E i 1 c δ ′ . If δ i 1 ′ = 0 , then

δ ′ = ( δ 1 ′ , … , 0 , … , δ r ′ ) ,

and we conjugate by c i 1 to obtain

E i 1 c i 1 ⁢ c δ ′ = E i 1 c δ ′ ⁢ c i 1 = E i 1 c δ ′ = E i 1 c δ ,

where we have used the facts that [ c i , c j ] = 1 and that c i 1 ∈ N G ⁢ ( E i 1 ) . So we deduce that δ ′′ = ( δ 1 ′ , … , 1 , … , δ r ′ ) also appears in the sum in 2 of this theorem. If δ i 1 ′ = 1 , then δ ′ = ( δ 1 ′ , … , 1 , … , δ r ′ ) , and we conjugate by c i 1 - 1 to obtain

E i 1 c i 1 - 1 ⁢ c δ ′ = E i 1 c δ ′ ⁢ c i 1 - 1 = E i 1 c δ ′ = E i 1 c δ ,

where we have used the facts that [ c i , c j ] = 1 and that c i 1 ∈ N G ⁢ ( E i 1 ) . So we deduce that δ ′′ = ( δ 1 ′ , … , 0 , … , δ r ′ ) also appears in the sum. ∎

Theorem 3.5

Let E = ⟨ e 1 , … , e r ⟩ be a maximal elementary abelian 𝑝-subgroup of 𝐺 of rank 𝑟, and assume that there are elements c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) with [ c i , c j ] = 1 for all 1 ≤ i , j ≤ r . Then H ~ r - 1 ⁢ ( | A p ⁢ ( G ) | ) ≠ 0 , and hence Quillen’s conjecture holds for 𝐺. If furthermore r = rk p ⁡ ( G ) , then Q ⁢ D p holds for 𝐺.

Proof

Note that the collection { E i , c j } is faithful by Lemma 2.6. Now apply Theorem 3.3 with A = Z , and Theorem 3.4 3 and 4 with

a δ = ( - 1 ) δ = ( - 1 ) δ 1 + ⋯ + δ r for ⁢ δ = ( δ 1 , … , δ r ) , δ i ∈ { 0 , 1 } . ∎

Definition 3.6

A collection satisfying the assumptions of Theorem 3.5 is called admissible, i.e., given a maximal elementary abelian 𝑝-subgroup E = ⟨ e 1 , … , e r ⟩ of 𝐺 with rk p ⁡ ( E ) = r , an admissible collection for 𝐺 is a collection { E i , c j } of subgroups E i = ⟨ e 1 , … , e i ^ , … , e r ⟩ of 𝐺 and elements c j of 𝐺 such that

c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) and [ c i , c j ] = 1 for all ⁢ 1 ≤ i , j ≤ r .

Note that such a collection is faithful by Lemma 2.6.

4 Applications to symmetric and classical groups

We start with an observation which is useful when investigating Quillen’s conjecture and Q ⁢ D p for p ′ -central quotients of finite groups.

Lemma 4.1

Let E = ⟨ e 1 , … , e r ⟩ be a maximal elementary abelian 𝑝-subgroup of 𝐺 of rank r ≤ rk p ⁡ ( G ) , and let c 1 , … , c r ∈ G . Let 𝑁 be a normal p ′ -subgroup of 𝐺 satisfying that, for each i ∈ S l r and all ϵ = ( ϵ 1 , … , ϵ r ) with ϵ i ∈ { - 1 , 0 , 1 } , we have E i c ϵ ∩ E ⁢ N ≤ E . In particular, this holds whenever 𝑁 is a normal p ′ -subgroup of 𝐺 which also centralizes 𝐸 (for instance, if 𝑁 is a central p ′ -subgroup of 𝐺). Set G ¯ = G / N , and denote by an overline ( * ) ¯ the image of elements or subgroups of 𝐺 in G ¯ . The following hold.

{ E i , c j } is faithful if and only if { E i ¯ , c j ¯ } is faithful.
c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) if and only if c i ¯ ∈ C G ¯ ⁢ ( E ¯ i ) ∖ N G ¯ ⁢ ( ⟨ e i ¯ ⟩ ) .

Proof

For part 1, assume first that { E i , c j } is faithful and that e ¯ i c ¯ ϵ ≤ E ¯ , i.e., that e i c ϵ ∈ E ⁢ N . Then, by assumption, we have e i c ϵ ∈ E , and as { E i , c j } is faithful, we get that c ϵ ∈ N G ⁢ ( ⟨ e i ⟩ ) . Hence we also get c ¯ ϵ ∈ N G ¯ ⁢ ( ⟨ e ¯ i ⟩ ) . Conversely, suppose that { E i ¯ , c j ¯ } is faithful. Suppose that e i c ϵ ∈ E . Then e ¯ i c ¯ ϵ ∈ E ¯ , and by assumption, c ¯ ϵ ∈ N G ¯ ⁢ ( ⟨ e ¯ i ⟩ ) . It follows that e i c ϵ ∈ ⟨ e i ⟩ ⁢ N ≤ E . By Dedekind’s modular law, ⟨ e i ⟩ ⁢ N ∩ E = ⟨ e i ⟩ ⁢ ( N ∩ E ) = ⟨ e i ⟩ . Thus { E i , c j } is faithful.

For part 2, assume that c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) for all 𝑖. Then c i ¯ ∈ C G ¯ ⁢ ( E i ¯ ) . We claim that c i ¯ ∉ N G ¯ ⁢ ( ⟨ e i ¯ ⟩ ) . Indeed, by assumption, if e i c i = e i a i ⁢ n for some n ∈ N and integer 0 < a i < p , then we have e i c i ∈ ⟨ e i ⟩ c i ∩ E N ≤ E , and since E ∩ N = { 1 } , we must have n = 1 and c i ∈ N G ⁢ ( ⟨ e i ⟩ ) .

Conversely, if c i ¯ ∉ N G ¯ ⁢ ( ⟨ e i ¯ ⟩ ) , then c i cannot normalize ⟨ e i ⟩ for any 1 ≤ i ≤ r . It remains to see that if c i ¯ ∈ C G ¯ ⁢ ( E ¯ i ) , then c i centralizes E i . For any j ≠ i , if e j c i ∈ e j ⁢ N , then by assumption, e j c i ∈ ⟨ e j ⟩ c i ∩ E N ≤ E , and so we must have e j c i = e j since E ∩ N = { 1 } . ∎

In the remainder of this section, we exhibit admissible collections for many alternating groups and finite classical groups of Lie type in non-defining characteristic, thereby proving that these groups possess the Quillen dimension at 𝑝 property (cf. Definition 3.6).

We write Sym X for the full permutation group on a finite set 𝑋. Instead, we write Sym n if X = { 1 , … , n } . Accordingly, we write A X and A n for the corresponding alternating groups.

Theorem 4.2

Suppose that p ≥ 5 , and let n ≥ p . Let G = A n or G = Sym n . Then there exists an admissible collection for 𝐺.

Proof

We refer the reader to [1, Chapter IV] for the 𝑝-local structure of the alternating and symmetric groups. The 𝑝-rank of 𝐺 is r = rk p ⁡ ( A n ) = ⌊ n p ⌋ . Consider the maximal elementary abelian 𝑝-subgroup 𝐸 of rank 𝑟 generated by

e 1 = ( 1 , … , p ) , … , e i = ( ( i - 1 ) ⁢ p + 1 , … , i ⁢ p ) , … , e r = ( ( r - 1 ) ⁢ p + 1 , … , r ⁢ p ) .

Define hyperplanes of 𝐸 by E i = ⟨ e 1 , e 2 , … , e ^ i , … , e r ⟩ . Write n = r ⁢ p + b with 0 ≤ b < p . Recall that | N Sym n ( ⟨ e i ⟩ ) : N A n ( ⟨ e i ⟩ ) | = | C Sym n ( E i ) : C A n ( E i ) | = 2 and that, since p ≥ 5 , the groups A n are simple for n ≥ p ; in particular, its Sylow 𝑝-subgroups are not normal ( A p contains ( p - 2 ) ! Sylow 𝑝-subgroups). We have

N G ⁢ ( ⟨ e i ⟩ ) = ( ⟨ e i , t i ⟩ × Sym { 1 , … , ( i - 1 ) ⁢ p , i ⁢ p + 1 , … , n } ) ∩ G , C G ⁢ ( E i ) = ( E i × Sym { ( i - 1 ) ⁢ p + 1 , … , i ⁢ p , r ⁢ p + 1 , … , n } ) ∩ G ,

where ⟨ e i , t i ⟩ ≅ C p ⋊ C p - 1 is the normalizer of ⟨ e i ⟩ in Sym { ( i - 1 ) ⁢ p + 1 , … , i ⁢ p } . Now, by assumption, r ≥ 1 and p ≥ 5 , and for each 1 ≤ i ≤ r , we choose

c i = ( ( i - 1 ) ⁢ p + 1 , ( i - 1 ) ⁢ p + 2 , ( i - 1 ) ⁢ p + 3 ) .

This element clearly belongs to C G ⁢ ( E i ) for G = A n and G = Sym n , and a short computation shows that it does not normalize ⟨ e i ⟩ . The collection { c i , 1 ≤ i ≤ r } satisfies the hypothesis of Theorem 3.5. Indeed, by definition, [ c i , c j ] = 1 for all i , j because the c i are all disjoint cycles. ∎

Remark 4.3

Computational evidence using GAP [13] shows that there is no admissible collection for p = 3 for the alternating and symmetric groups of degree 𝑛 unless n = 4 , 5 or 8. For n = 4 , 5 , these groups have 3-rank 1 and no non-trivial normal 3-subgroup. Then it is immediate that there exists an admissible collection. For A 8 , we can take E = ⟨ e 1 , e 2 ⟩ with e 1 = ( 1 , 2 , 3 ) and e 2 = ( 4 , 5 , 6 ) , and put c 1 = ( 1 , 7 ) ⁢ ( 2 , 3 ) and c 2 = ( 4 , 8 ) ⁢ ( 5 , 6 ) . Then a direct check shows that { E i , c j } is an admissible collection for A 8 (and so for Sym 8 too). Note that these cases are known via other methods [10, Proposition 2.10] since A n has 3-rank at most 2 for n = 4 , 5 , 8 and p = 3 .

We will now adapt the above argument to find admissible collections in classical finite groups with trivial 𝑝-core in non-defining characteristic, and therefore show that they have the Quillen dimension at 𝑝 property. By a classical group, we mean 𝐺 a finite group of Lie type A n , A n 2 , B n , C n , D n or D n 2 , and we refer the reader to [7, Table 22.1] for the different isogeny types and the standard notation that we will use. In particular, if 𝐺 is of type A n - 1 defined over a field with 𝑞 elements, then 𝐺 is a central quotient of a group G * such that SL n ⁡ ( q ) ≤ G * ≤ GL n ⁡ ( q ) , and G = G * / Z for a central subgroup Z ≤ Z ⁢ ( G * ) . If Z = Z ⁢ ( G * ) , then we write P ⁢ G * = G * / Z ⁢ ( G * ) . We also assume that 𝑝 is coprime to the characteristic of the field of definition of 𝐺.

Notation 4.4

For any natural number 𝑛, we write

GL n + ⁡ ( q ) = GL n ⁡ ( q ) , GL n - ⁡ ( q ) = GU n ⁡ ( q ) , SL n + ⁡ ( q ) = SL n ⁡ ( q ) , SL n - ⁡ ( q ) = SU n ⁡ ( q )

for linear, unitary, special linear and special unitary groups respectively. So a superscript “+” means “linear” and “ - ” means “unitary”. We let 𝑑 be the multiplicative order of 𝑞 modulo 𝑝, that is, the smallest positive integer such that 𝑝 divides q d - 1 . Finally, if 𝑛 is an integer, we write ( n ) p for the 𝑝-part of 𝑛.

For the central quotient P ⁢ G = G / Z ⁢ ( G ) of a classical group 𝐺, recall that

O p ⁢ ( G ) ≤ Z ⁢ ( G )

except for some small linear and unitary groups. Moreover, we have isomorphisms Z ⁢ ( GL n ϵ ⁡ ( q ) ) ≅ C q - ϵ and Z ⁢ ( SL n ϵ ⁡ ( q ) ) ≅ C ( n , q - ϵ ) for ϵ = ± 1 , and the centers of symplectic and orthogonal groups are 2-groups (cf. [7, Table 24.2]). Hence, P ⁢ G is a p ′ -quotient of 𝐺 if and only if O p ⁢ ( G ) = 1 . If so, P ⁢ G and 𝐺 have the same 𝑝-subgroup structure [2, Lemma 0.11], in particular, A p ⁢ ( G ) ≅ A p ⁢ ( P ⁢ G ) . In this case, we show that Q ⁢ D p holds for both groups by exhibiting an admissible collection for 𝐺, and then applying Lemma 4.1, we obtain that P ⁢ G possesses an admissible collection too.

In the remaining case, i.e., if O p ⁢ ( G ) > 1 , we show that there is no admissible collection for 𝐺, using Lemma 2.7, and we prove that Q ⁢ D p holds for P ⁢ G by exhibiting an admissible collection. Table 1 summarizes the different cases, indicating the conditions and the results to be applied in each case. For conciseness, we omitted from the table the exceptions ( p , q ) = ( 3 , 2 ) , which apply whenever 𝐺 is not a linear group, and the cases PSL 3 ⁡ ( q ) with q ≡ 1 ( mod 3 ) at the prime 3.

Table 1

Group 𝐺	O p ⁢ ( G ) = 1		O p ⁢ ( G ) > 1
	d > 1		d = 1

GL n ⁡ ( q )	Q ⁢ D p by 4.7, 4.10		∄ admissible coll. by 4.5
PGL n ⁡ ( q )	Q ⁢ D p by 4.1		Q ⁢ D p by 4.9

	d > 1	d = 1 , ( p , n ) = 1	d = 1 , ( p , n ) = p

SL n ⁡ ( q )	Q ⁢ D p by 4.7, 4.10	Q ⁢ D p by 4.8	∄ admissible coll. by 4.5
PSL n ⁡ ( q )	Q ⁢ D p by 4.1	Q ⁢ D p by 4.1	Q ⁢ D p by 4.9

	d ≠ 2		d = 2

GU n ⁡ ( q )	Q ⁢ D p by 4.10		∄ admissible coll. by 4.5
PGU n ⁡ ( q )	Q ⁢ D p by 4.1

	d ≠ 2		d = 2 , ( p , n ) = p

SU n ⁡ ( q )	Q ⁢ D p by 4.10		∄ admissible coll. by 4.5
PSU n ⁡ ( q )	Q ⁢ D p by 4.1

symplectic	Q ⁢ D p for 𝐺 by 4.10
orthogonal	Q ⁢ D p for P ⁢ G by 4.1		—

Note that d = 1 if and only if q ≡ 1 ( mod p ) , and that d = 2 if and only if q ≡ - 1 ( mod p ) , which determines when a linear or unitary group has a non-trivial 𝑝-core. The last two rows denote any symplectic or orthogonal finite group of Lie type, as described in [7, Table 22.1].

Proposition 4.5

Suppose that 𝑝 is odd and that n ≥ 1 . Let

G = GL n ϵ ⁡ ( q ) or G = SL n ϵ ⁡ ( q ) ,

and assume that q - ϵ ≡ 0 ( mod p ) or ( q - ϵ , n ) ≡ 0 ( mod p ) respectively. Then there exists no admissible collection for 𝐺.

Proof

Recall that Z ⁢ ( GL n ϵ ⁡ ( q ) ) is cyclic of order q - ϵ and Z ⁢ ( SL n ϵ ⁡ ( q ) ) is cyclic of order ( q - ϵ , n ) . So the assumptions say that O p ⁢ ( G ) ∩ Z ⁢ ( G ) > 1 , and the assertion follows from Lemma 2.7. ∎

We now consider the linear groups with d > 1 .

Notation 4.6

We denote by I n the n × n identity matrix in M n × n ⁢ ( F q ) and by M a ⁢ b the standard basis matrix in M n × n ⁢ ( F q ) , i.e., the n × n matrix with coefficients in F q whose entry at row 𝑐 and column 𝑑 is given by

( M a ⁢ b ) c ⁢ d = { 1 if ⁢ a = c ⁢ and ⁢ b = d , 0 otherwise .

Proposition 4.7

Suppose that 𝑝 is odd, and let G = GL n ⁡ ( q ) or G = SL n ⁡ ( q ) with d > 1 except SL 2 ⁡ ( 2 ) for p = 3 . Then there exists an admissible collection for 𝐺.

Proof

Write n = r ⁢ d + f , with 0 ≤ f < d . Then the unique (up to 𝐺-conjugacy) maximal elementary abelian 𝑝-subgroup 𝐸 of 𝐺 has rank 𝑟. We split the proof into two cases. First, suppose that ( p , q ) ≠ ( 3 , 2 ) . Using [5, Section 4.10, 4.8] and the fact that GL 1 ⁡ ( q d ) ↪ GL d ⁡ ( q ) , we can choose E ≤ G as the set of block diagonal matrices of the form E = ⟨ θ 1 ⁢ ( u ) , … , θ r ⁢ ( u ) ⟩ , where u ∈ GL 1 ⁡ ( q d ) has order 𝑝, and where

θ i : GL 1 ⁡ ( q d ) → GL d ⁡ ( q ) → GL n ⁡ ( q ) , x ↦ X ↦ diag ⁡ ( I ( i - 1 ) ⁢ d , X , I n - i ⁢ d ) .

Put e j = θ i ⁢ ( u ) for 1 ≤ i ≤ r . Note that det ⁡ ( e i ) = 1 because det ⁡ ( e i ) must be a 𝑝-th root of 1 in F q and ( p , q - 1 ) = 1 . Then

C G ⁢ ( E ) = G ∩ ( diag ⁡ ( Im ⁡ ( θ 1 ) , … , Im ⁡ ( θ r ) ) × GL f ⁡ ( q ) )

with Im ⁡ ( θ i ) ≅ GL 1 ⁡ ( q d ) for all 1 ≤ i ≤ r . We have

C G ⁢ ( E i ) ≅ G ∩ ( C GL n ⁡ ( q ) ⁢ ( E ) ⁢ GL d + f ⁡ ( q ) ) , N G ⁢ ( ⟨ e i ⟩ ) ≅ G ∩ ( ( GL 1 ⁡ ( q d ) ⋊ C d ) × GL n - d ⁡ ( q ) ) ,

where the factors GL d + f ⁡ ( q ) and GL n - d ⁡ ( q ) intersect in a subgroup isomorphic to GL f ⁡ ( q ) . So we can choose c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) in the factor GL d ⁡ ( q ) properly containing the subgroup ( GL 1 ⁡ ( q d ) ⋊ C d ) in the 𝑖-th diagonal block for all 𝑖. Such a choice is possible whenever 𝑝 is odd, unless q = 2 , d = 2 and p = 3 , because

GL 1 ⁡ ( 2 2 ) ⋊ C 2 ≅ GL 2 ⁡ ( 2 ) ≅ Sym 3 .

Now, our choice satisfies c i ∈ C G ⁢ ( E i ) ∖ N G ⁢ ( ⟨ e i ⟩ ) and [ c j , c i ] = 1 for all i , j . Thus { E i , c j } is an admissible collection for 𝐺.

It remains to handle the case ( p , q ) = ( 3 , 2 ) . If n = 2 , then O 3 ⁢ ( SL 2 ⁡ ( 2 ) ) ≠ 1 , and this case is excluded by assumption. If n = 3 , recall that a Sylow 3-subgroup of G = SL 3 ⁡ ( 2 ) has order 3 and it is not normal in 𝐺, saying that, trivially, there exists an admissible collection for 𝐺. If n ≥ 4 , we observe that it is enough to consider the groups G = SL 2 ⁢ k ⁡ ( 2 ) , for k ≥ 2 , as the index of SL 2 ⁢ k ⁡ ( 2 ) in SL 2 ⁢ k + 1 ⁡ ( 2 ) is not divisible by 3 so that an admissible collection for SL 2 ⁢ k ⁡ ( 2 ) induces an admissible collection for SL 2 ⁢ k + 1 ⁡ ( 2 ) . As above, we can choose an elementary abelian 3-subgroup 𝐸 of 𝐺 of maximal rank embedded in the subgroup formed by the diagonal 2 × 2 blocks in SL 2 ⁢ k ⁡ ( 2 ) . Moreover, we see that it suffices to find admissible collections with respect to such an 𝐸 for G = SL 4 ⁡ ( 2 ) and for G = SL 6 ⁡ ( 2 ) because, from them, we can obtain admissible collections for SL 2 ⁢ n ⁡ ( 2 ) for all n ≥ 2 . Using GAP [13], we find the following admissible collections. Let X = ( 0 1 1 1 ) and Y = ( 0 1 1 0 ) . If G = SL 4 ⁡ ( 2 ) , put

e 1 = ( X 2 X 2 ) , e 2 = ( X 2 X ) , c 1 = ( X Y X ) , c 2 = ( X X X 2 ) ,

where a blank entry means the zero matrix of appropriate size. A direct calculation shows that E = ⟨ e 1 , e 2 ⟩ = C 3 × C 3 is an elementary abelian 3-subgroup of 𝐺 of maximal rank, with c j ∈ C G ⁢ ( e 3 - j ) and c j ∉ N G ⁢ ( ⟨ e j ⟩ ) for j = 1 , 2 , and also that [ c 1 , c 2 ] = 1 . Thus these elements form an admissible collection for SL 4 ⁡ ( 2 ) . If G = SL 6 ⁡ ( 2 ) , put

e 1 = ( X 2 I 2 I 2 ) , e 2 = ( X X X ) , e 3 = ( X 2 X 2 X )

and

c 1 = ( X X I 2 I 2 ) , c 2 = ( X X Y X ) , c 2 = ( X X X X 2 ) .

A direct computation shows that E = ⟨ e 1 , e 2 , e 3 ⟩ = C 3 × C 3 × C 3 is an elementary abelian 3-subgroup of 𝐺 of maximal rank, with

c i ∈ C G ⁢ ( ⟨ e j , e k ⟩ ) and c i ∉ N G ⁢ ( ⟨ e i ⟩ ) for any ⁢ { i , j , k } = { 1 , 2 , 3 } ,

and also that [ c i , c j ] = 1 for any 1 ≤ i , j ≤ 3 . Thus these elements form an admissible collection for SL 6 ⁡ ( 2 ) . ∎

The cases G = SL n ⁡ ( q ) , PSL n ⁡ ( q ) and PGL n ⁡ ( q ) when d = 1 are more subtle. In this case, O p ⁢ ( SL n ⁡ ( q ) ) = 1 if and only if ( n , p ) = 1 , i.e. when SL n ⁡ ( q ) has no scalar matrix ζ ⁢ I n , where ζ n = ζ p = 1 ≠ ζ . From [5, Section 4.10], the 𝑝-ranks of SL n ⁡ ( q ) , PSL n ⁡ ( q ) and PGL n ⁡ ( q ) may be found, and we use this information in the proofs of the next two propositions.

Proposition 4.8

Let G = SL n ⁡ ( q ) with 𝑝 an odd prime dividing q - 1 and coprime to 𝑛. Then there exists an admissible collection for 𝐺.

Proof

By assumption, O p ⁢ ( G ) = 1 because 𝑝 does not divide 𝑛. Let u ∈ GL 1 ⁡ ( q ) be an element of order 𝑝, and put, for all 1 ≤ i ≤ n - 1 ,

e i = diag ⁡ ( u ⁢ I i - 1 , u 1 - n , u ⁢ I n - i ) .

Let E = ⟨ e 1 , … , e n - 1 ⟩ so that E ∈ A p ⁢ ( G ) has maximal rank n - 1 = rk ⁡ ( G ) . Then C G ( E ) = T = C G ( T ) ≅ GL 1 ( q ) n - 1 is the subgroup of 𝐺 formed by all diagonal matrices with determinant 1. We have

C G ⁢ ( E i ) ≅ G ∩ ( T ⁢ GL 2 ⁡ ( q ) ) and N G ⁢ ( ⟨ e i ⟩ ) = C G ⁢ ( e i ) ≅ GL n - 1 ⁡ ( q ) ,

where such GL n - 1 ⁡ ( q ) contains 𝑇 and where

C G ⁢ ( E i ) ∩ N G ⁢ ( ⟨ e i ⟩ ) = T .

Choose c j = I n + M j ⁢ n (see Notation 4.6). A routine calculation shows that

c j ∈ C G ⁢ ( E j ) ∖ N G ⁢ ( ⟨ e j ⟩ )

has order 𝑞 and that [ c j , c i ] = 1 for all i , j . Thus { E i , c j } is an admissible collection for 𝐺. ∎

Proposition 4.9

Let G = PSL n ⁡ ( q ) or PGL n ⁡ ( q ) with 𝑝 an odd prime dividing ( n , q - 1 ) or q - 1 respectively, excluding the case PSL 3 ⁡ ( q ) for p = n = 3 . Then there exists an admissible collection for 𝐺.

Proof

First consider G = PSL n ⁡ ( q ) . If p = n = 3 , then the 𝑝-rank is 2, and we exclude this case by assumption. Otherwise, from [4, Theorem 10.6 (1)], we have

rk p ⁡ ( G ) = { n - 2 if ⁢ ( n ) p ≥ ( q - 1 ) p , n - 1 if ⁢ ( n ) p < ( q - 1 ) p .

For the former case, we embed SL n - 1 ⁡ ( q ) in PSL n ⁡ ( q ) as the subgroup

( SL n - 1 ⁡ ( q ) 0 ⋮ 0 0 … 0 1 ) ,

and since 𝑝 does not divide n - 1 , we invoke Proposition 4.8 to construct an admissible collection with the maximal elementary abelian 𝑝-subgroup 𝐸 of 𝐺 of rank n - 2 and c 1 , … , c n - 2 all sitting in SL n - 1 ⁡ ( q ) .

For the case G = PSL n ⁡ ( q ) and ( n ) p < ( q - 1 ) p , let z , u ∈ F q * be elements such that 𝑧 has order ( n ) p and u p = z . Set

e = diag ⁡ ( z , … , z ) ∈ Z ⁢ ( SL n ⁡ ( q ) ) ,

and as in Proposition 4.8, set, for all 1 ≤ i ≤ n - 1 ,

e i = diag ⁡ ( u ⁢ I i - 1 , u 1 - n , u ⁢ I n - i ) ∈ SL n ⁡ ( q ) .

Then e i p = e , and in the quotient G = SL n ⁡ ( q ) / Z ⁢ ( SL n ⁡ ( q ) ) , the classes of the elements e 1 , … , e n - 1 generate a subgroup E ∈ A p ⁢ ( G ) of rank n - 1 = rk p ⁡ ( G ) . Choose c j to be the class of the element I n + M j ⁢ n ∈ SL n ⁡ ( q ) (see Notation 4.6). A routine calculation shows that we have built an admissible collection for 𝐺.

Next let G = PGL n ⁡ ( q ) . Then we have rk p ⁡ ( G ) = n - 1 . We choose elements e 1 , … , e n - 1 as follows: let u ∈ GL 1 ⁡ ( q ) be an element of order 𝑝, and set

e i = diag ⁡ ( u ⁢ I i - 1 , 1 , u ⁢ I n - i ) for ⁢ 1 ≤ i ≤ n - 1 .

Let E = ⟨ e 1 , … , e n - 1 ⟩ , and define c j = I n + M j ⁢ n (see Notation 4.6). It is clear that c j ∈ C G ⁢ ( E j ) and that [ c i , c j ] = 1 for all 1 ≤ j ≤ n - 1 . Moreover, a routine calculation shows that the matrices e j c j are not diagonal, and hence c j ∉ N G ⁢ ( ⟨ e j ⟩ ) and { E i , c j } is an admissible collection for 𝐺. ∎

Machine computations indicate that the cases excluded in Proposition 4.9 are probably genuine exceptions to the existence of admissible collections.

We are now ready to handle the remaining classical groups (cf. the paragraph above Notation 4.4 for our conventions). We follow the methods in [4, Part I, Section 8], and so let 𝐺 act on a vector space 𝑉, defined over F q , unless 𝐺 is unitary, in which case 𝑉 is defined over F q 2 , and 𝑉 comes equipped with a hermitian form. Let Isom ⁢ ( V ) denote the full isometry group of 𝑉. So, for instance, if 𝐺 has Lie type A n - 1 ⁡ ( q ) , then Isom ⁢ ( V ) = GL n ⁡ ( q ) , and 𝐺 is a central quotient of a group G * such that SL n ⁡ ( q ) ≤ G * ≤ GL n ⁡ ( q ) .

Theorem 4.10

Let 𝐺 be a finite linear, unitary, symplectic or orthogonal group defined over a field with 𝑞 elements (respectively q 2 elements if 𝐺 is unitary), and let 𝑝 be an odd prime with p ∤ q . Suppose that the following conditions hold.

𝑝 divides | G | and O p ⁢ ( G ) = 1 .
( p , q ) ≠ ( 3 , 2 ) unless 𝐺 is linear.
p ∤ q + 1 if 𝐺 is unitary.

Then there exists an admissible collection for 𝐺.

Remark 4.11

Some well-known properties of small finite groups of Lie type (cf. [7, Remark 24.18] for instance) and computational evidence using GAP [13] show that there are no admissible collections for small dimensional unitary, symplectic and orthogonal groups for p = 3 and q = 2 . So condition (ii) in the statement is necessary. The case when 𝐺 is linear and ( p , q ) = ( 3 , 2 ) is dealt with in Proposition 4.7.

Proof

From Propositions 4.8 and 4.9, we can assume that if 𝐺 is linear, then d > 1 . Condition (iii) is equivalent to saying that if 𝐺 is unitary, then d ≠ 2 .

Let 𝑉 be the underlying vector space of 𝐺, and Isom ⁢ ( V ) its full isometry group, as explained above. From [7, Tables 22.1 and 24.2], our assumptions allow us to suppose that 𝐺 is simply connected. Indeed, the index of the group of simply connected type in Isom ⁢ ( V ) is coprime to 𝑝 so that both have isomorphic Sylow 𝑝-subgroups. Thus, if there is an admissible collection for the group of simply connected type, then its inclusion in Isom ⁢ ( V ) yields an admissible collection for Isom ⁢ ( V ) . Hence, if 𝐺 is a central quotient of G * , the image of such an admissible collection is an admissible collection for 𝐺, by Lemma 4.1. For instance, if 𝐺 has Lie type A n - 1 ⁡ ( q ) , then we can suppose that G = SL n ⁡ ( q ) . Indeed, if d > 1 (i.e. if q ≢ 1 ( mod p ) ), then an admissible collection for SL n ⁡ ( q ) gives admissible collections for any group G * / Z , where SL n ⁡ ( q ) ≤ G * ≤ GL n ⁡ ( q ) and Z ≤ Z ⁢ ( G * ) .

Let 𝐸 be a maximal elementary abelian 𝑝-subgroup of 𝐺 of maximal order. Choose generators { e 1 , … , e r } of 𝐸 and a basis { v 1 , … , v n } of 𝑉 such that e i acts as the identity everywhere except on V i = ⟨ v ( i - 1 ) ⁢ d + 1 , … , v i ⁢ d ⟩ for all 1 ≤ i ≤ r . Put V 0 = ⟨ v r ⁢ d + 1 , … , v n ⟩ , possibly V 0 = { 0 } . Thus V = ⊕ 0 ≤ i ≤ r V i . We refer the reader to [4, Table 10:1] for the values of 𝑟 depending on the type of 𝐺, and the values of 𝑛 and 𝑑. (Recall that, under our assumptions, 𝐺 and Isom ⁢ ( V ) have isomorphic Sylow 𝑝-subgroups.)

From [4, Part I, Section 8, (8-1)], we have, for all 1 ≤ i ≤ r ,

C G ⁢ ( e i ) = G ∩ ( Isom ⁢ ( V / V i ) × GL 1 ϵ ⁡ ( q e ) ) .

It follows that

C G ⁢ ( E i ) = ⋂ j ≠ i C G ⁢ ( e j ) = G ∩ ( GL 1 ϵ ( q e ) ( 1 ) × ⋯ × GL 1 ϵ ( q e ) ( i - 1 ) × Isom ⁢ ( V i ⊕ V 0 ) × GL 1 ϵ ( q e ) ( i + 1 ) × ⋯ × GL 1 ϵ ( q e ) ( r ) )

for 1 ≤ i ≤ r , and

C G ( E ) = G ∩ ( GL 1 ϵ ( q e ) ( 1 ) × ⋯ × GL 1 ϵ ( q e ) ( r ) × Isom ( V 0 ) ) ,

where GL 1 ϵ ( q e ) ( i ) = C Isom ⁢ ( V i ) ( e i | V i ) is the centralizer of the action of e i restricted to V i , where the sign ϵ = ± 1 and the value of 𝑒 depend on the parity of 𝑑 and the type of 𝐺, and where Isom ⁢ ( V 0 ) has the same type as 𝐺. Explicitly, we have the following.

If 𝐺 is linear, then ϵ = + and d = e .
If 𝐺 is unitary, then ϵ = - if d ≡ 2 ( mod 4 ) and ϵ = + otherwise. We put e = 2 ⁢ d if 𝑑 is odd, e = d 2 if d ≡ 1 ( mod 4 ) and e = d if d ≡ 0 ( mod 4 ) .
If 𝐺 is symplectic or orthogonal, put f = 1 2 ⁢ lcm ⁡ ( 2 , d ) and hence ϵ = + if 𝑓 is odd and - otherwise, and put e = 2 ⁢ f . (We refer the reader to [4, Part I, Section 8] for the type of the orthogonal space, as it does not impact on our argument.)

By definition, dim ⁡ ( V 0 ) < d (respectively 2 ⁢ d if 𝐺 is symplectic or orthogonal), saying that p ∤ | Isom ⁢ ( V 0 ) | .

Also,

N G ( ⟨ e i ⟩ ) = G ∩ ( ( GL 1 ϵ ( q e ) ( i ) ⋊ C e ( i ) ) × Isom ( ⊕ j ≠ i V j ) ) ,

where 𝑒 is as above.

Conditions (i)–(iii) in the statement ensure that

( GL 1 ϵ ( q d ) ( j ) ⋊ C e ( j ) ) ⪇ Isom ( V j )

so that we can pick c j ∈ G such that c j acts as the identity on V i for all i ≠ j and the restriction c j | V j ∈ Isom ( V j ) ∖ ( GL 1 ϵ ( q d ) ( j ) ⋊ C e ( j ) ) .

Since V = ⊕ 0 ≤ i ≤ r V i , we have [ Isom ⁢ ( V i ) , Isom ⁢ ( V j ) ] = 1 for all i ≠ j , and therefore the elements c j commute pairwise, i.e. [ c i , c j ] = 1 for any 1 ≤ i , j ≤ r . We conclude that { E i , c j } is an admissible collection for 𝐺. ∎

Our objective in this work was to devise a simpler argument to show the Quillen dimension at 𝑝 property for the symmetric and alternating groups, and for the finite classical groups in non-defining characteristic. Although our results do not fully meet our initial objective, we believe that our methods can be further generalized to tackle the cases excluded from the present paper, and also 𝑝-extensions, i.e., almost-simple groups with an elementary abelian 𝑝-group inducing outer automorphisms. The first author will pursue this aim in a subsequent work.

Funding source: Junta de Andalucía

Award Identifier / Grant number: FQM-213

Funding statement: First author supported by MEC grant MTM2016-78647-P and Junta de Andalucía grant FQM-213.

Acknowledgements

The authors are sincerely grateful to Gunter Malle for his helpful suggestions in this research work.

Communicated by: Evgeny Vdovin

References

[1] A. Adem and R. J. Milgram, Cohomology of Finite Groups, 2nd ed., Grundlehren Math. Wiss. 309, Springer, Berlin, 2004. 10.1007/978-3-662-06280-7Search in Google Scholar

[2] M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex, Ann. of Math. (2) 137 (1993), no. 3, 473–529. 10.2307/2946530Search in Google Scholar

[3] A. Díaz Ramos, On Quillen’s conjecture for 𝑝-solvable groups, J. Algebra 513 (2018), 246–264. 10.1016/j.jalgebra.2018.07.027Search in Google Scholar

[4] D. Gorenstein and R. Lyons, The local structure of finite groups of characteristic 2 type, Mem. Amer. Math. Soc. 42 (1983), No. 276. 10.1090/memo/0276Search in Google Scholar

[5] D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Math. Surveys Monogr. 40, American Mathematical Society, Providence, 1998. 10.1090/surv/040.4Search in Google Scholar

[6] T. Hawkes and I. M. Isaacs, On the poset of 𝑝-subgroups of a 𝑝-solvable group, J. Lond. Math. Soc. (2) 38 (1988), no. 1, 77–86. 10.1112/jlms/s2-38.1.77Search in Google Scholar

[7] G. Malle and D. Testerman, Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Stud. Adv. Math. 133, Cambridge University Press, Cambridge, 2011. 10.1017/CBO9780511994777Search in Google Scholar

[8] K. I. Piterman, An approach to Quillen’s conjecture via centralisers of simple groups, Forum Math. Sigma 9 (2021), Paper No. e48. 10.1017/fms.2021.41Search in Google Scholar

[9] K. I. Piterman, I. Sadofschi Costa and A. Viruel, Acyclic 2-dimensional complexes and Quillen’s conjecture, Publ. Mat. 65 (2021), no. 1, 129–140. 10.5565/PUBLMAT6512104Search in Google Scholar

[10] D. Quillen, Homotopy properties of the poset of nontrivial 𝑝-subgroups of a group, Adv. Math. 28 (1978), no. 2, 101–128. 10.1016/0001-8708(78)90058-0Search in Google Scholar

[11] J. J. Rotman, An Introduction to Algebraic Topology, Grad. Texts in Math. 119, Springer, New York, 1988. 10.1007/978-1-4612-4576-6Search in Google Scholar

[12] S. D. Smith, Subgroup Complexes, Math. Surveys Monogr. 179, American Mathematical Society, Providence, 2011. 10.1090/surv/179Search in Google Scholar

[13] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.10.1, 2019, (https://www.gap-system.org). 10.1093/oso/9780190867522.003.0002Search in Google Scholar

Received: 2021-03-02

Revised: 2021-07-29

Published Online: 2021-09-16

Published in Print: 2022-01-01

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