arXiv:1604.01922v1 [math.AT] 7 Apr 2016

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS
ANTONIO DÍAZ RAMOS
Abstract. We give a new proof of Quillen’s conjecture for solvable groups
via a geometric and explicit method. For p-solvable groups, we provide both a
new proof using the Classification of Finite Simple Groups and an asymptotic
version without employing it.

1. Introduction
Let G be a finite group, let p be a prime and let Ap (G) be the poset consisting
of the non-trivial elementary abelian p-groups of G ordered by inclusion. The
homotopy properties of the topological realization |Ap (G)| were first studied in [8].
There, Quillen introduced the following conjecture, where we denote by Op (G) the
largest normal p-subgroup of G:
Conjecture 1.1 (Quillen’s conjecture). If |Ap (G)| is contractible then Op (G) 6= 1.
We start discussing some of the known progress on this conjecture as well as we
briefly comment on the methods employed so far. If the implication 1.1 holds for
G we say that G satisfies QC. Let us introduce the following notion.
Definition 1.2. Let G be a finite group of p-rank r. We say that G has Quillen
e r−1 (|Ap (G)|; Q) 6= 0.
dimension at p if Op (G) = 1 ⇒ H

This notion was introduced by Ashchbacher and Smith in [2, p.474] and we
e ∗ (|Ap (G)|; Q)
denote it by QDp . Note that r − 1 is the top dimension for which H
can possibly be non-zero. As contractibility leads to zero homology it is clear that:
QDp holds for G⇒ QC holds for G.

Quillen observed the following:
(1)

QDp for all K ⋊ Frp with K solvable p′ -group ⇒ QDp for all G solvable,

where Frp is an elementary abelian group of order pr acting on K and K ⋊Frp is their
semidirect product. In fact, the left hand side of the implication above was proven
by Quillen himself [8, Corollary 12.2], leading to the solution of the conjecture
in the solvable case. This is an inductive proof built on Cohen-Macaulay posets.
Alperin supplied an alternative approach that examines a minimal counterexample
via coprime action results [9, Theorem 8.2.9]. A further observation of Quillen is:
(2)

QDp for all K ⋊ Frp with K p′ -group ⇒ QDp for all G p-solvable.

Note that the solvability requirement on K has been dropped. In this case, the
left hand side, and hence QC for p-solvable groups, holds by an extension of the
Date: January 18, 2018.
Supported by MICINN grant RYC-2010-05663. Partially supported by MEC grant MTM201341768-P and Junta de Andalucı́a grant FQM-213.
1

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ANTONIO DÍAZ RAMOS

aforementioned argument of Alperin via the Classification of the Finite Simple
Groups (CFSG). See [3, Theorem 1], [9, Theorem 8.2.12] and [2, Theorem 0.5].
As contractiblity produces zero homology with any trivial coefficients, it makes
sense to include in the above discussion homology with (trivial) coefficients in any
abelian group A. In this work, we investigate certain homology classes that we
e c (|Ap (K ⋊ Fr )|; A) of
term constructible classes and that form a subgroup H
r−1
p
e r−1 (|Ap (K ⋊ Fr )|; A), see Definition 4.3 for details.
H
p
Definition 1.3. Consider the semidirect product K ⋊ Frp with K a p′ -group and let
A be any abelian group. We say that K ⋊ Frp has constructible Quillen dimension
c
e r−1
at p and A if Op (K ⋊ Frp ) = 1 ⇒ H
(|Ap (K ⋊ Frp )|; A) 6= 0.

We denote this condition by QDpc,A . By the expression QDpc we mean that QDpc,A
holds for some abelian group A. Note that for the semidirect product K ⋊ Frp :
Op (K ⋊ Frp ) = 1 ⇔ the action of Frp on K is faithful.

(3)

As before, we have that:
(4)

QDpc for all K ⋊ Frp with K solvable p′ -group ⇒ QC for all G solvable

and
(5)

QDpc for all K ⋊ Frp with K p′ -group ⇒ QC for all G p-solvable.

A constructible class for the group K ⋊ Frp is defined by a choice of an element
of A for each Sylow p-subgroup of this group, a· = (aS )S∈Sylp (K⋊Frp ) , such that
certain homogeneous linear equations are satisfied. Non-trivial solutions of these
c
e r−1
equations gives rise to non-trivial elements in H
(|Ap (K ⋊ Frp )|; A). So, roughly
speaking, we have removed a subdivision when computing constructible homology
classes. This means that we only need to look at the top two layers of this poset,
i.e., Sylow subgroups and their hyperplanes. We show below that constructible
classes suffice to prove Quillen’s conjecture in the p-solvable case. We start with
the following findings.
Theorem 1.4. For the elementary abelian group Frp or order pr we have:
c,Z

(a) QDp q holds for K ⋊ Frp with K a q-group and q 6= p.
(b) QDpc,Z2 holds for K ⋊ Frp with K an abelian p′ -group.
Here, ZD = Z/DZ. We succinctly outline the families a· utilised for this result.
For case (a), we choose a· to take the constant value 1 ∈ Zq . Letting a· to be the
characteristic function of an appropriately selected subset of Sylow p-subgroups
gives (b). For the next result, we show that the aforementioned linear equations
have at least a non-trivial solution.
Theorem 1.5. QDpc,Z holds for K ⋊ Frp with K a p′ -group if |K| = q1 e1 · · · ql el
satisfies that r < qi for all i = 1, . . . , l.
In view of the previous discussion, Theorem 1.5 is in fact an asymptotic version
of Quillen’s conjecture for p-solvable groups. If K is solvable, its Fitting subgroup
is self-centralizing and has abelian Frattini quotient. These two techniques are
of interest here because they preserve faithfulness and because of Equation (3).
Hence, the solvable case can be studied via the abelian case above together with
the following fact that constructible classes behave extremely well with respect to
quotients.

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

3

Theorem 1.6. Consider K ⋊Frp with K a p′ -group and let N EK be an Frp -invariant
e c (|Ap (K/N ⋊ Fr )|; A) 6= 0 then
subgroup of K. Let A be any abelian group. If H
r−1
p
c
r
e
)|;
A)
=
6
0.
(|A
(K
⋊
F
H
p
p
r−1
From here and Theorem 1.4(b) we obtain the next result.

Theorem 1.7. QDpc,Z2 holds for K ⋊ Frp with K a solvable p′ -group.
This result together with Equation (4) has the following immediate consequence.
Theorem 1.8. Quillen’s conjecture holds for solvable groups.
For general p′ -group K, the Frattini quotient of its generalized Fitting subgroup
is the direct product of an abelian group with simple groups. Theorem 1.6 again
reduces the problem to building constructible classes on this quotient. As in Theorem 1.4(b), we choose a· to be the characteristic function of a certain subset of
Sylow p-subgroups, and we use the CFSG to single out this subset.
Theorem 1.9. QDpc,Z2 holds for K ⋊ Frp with K a p′ -group.
Now employing Equation (5) we obtain the p-solvable case.
Theorem 1.10. Quillen’s conjecture holds for p-solvable groups.
Going back to the initial discussion, we point out that Alperin’s arguments for
the solvable and p-solvable cases are aimed to build a homology sphere in top
dimension r − 1. This sphere is built by appropriately choosing points to form r 0spheres, and the considering its join. Here, our constructible homology class in top
dimension r − 1 is made out of conical pieces of that dimension glued along a graph,
see Examples 3.4, 4.9 and 5.7. The weight of our arguments lies in the geometry
of the situation, and we do not use neither induction nor minimal counterexample
reasoning. For more details on this comparison, please see Remarks 5.4 and 6.8.
The methods explained here are open in at least a couple of directions: First, one
e c (|Ap (G)|; A) for any finite group
can also define the constructible subgroup H
r−1
G of rank r and any abelian group A. Nevertheless, the situation is more complicated as, to start with, the coprime action Propositions 2.2 and 2.3 do not hold in
general. Second, it is easy to sharpen Theorem 5.3 in several ways. For instance,
the conditions ci normalizes CK (H) and [ci , cj ] ∈ CK (H) for all i and j, are also
enough to have a 2-system.
Remark 1.11. By [8, Proposition 2.4], which is the reversed implication of Cone ∗ (|Ap (K ⋊ Frp )|; A) 6= 0 forces the action of Frp on K
jecture 1.1, the condition H
to be faithful. In particular, if the thesis of Theorem 5.6 holds, we deduce that Frp
acts faithfully on K and on K/N .
Acknowledgements: I am really grateful to Stephen D. Smith for his time and
generosity. He read an earlier version of this work and contributed many interesting
and clarifying comments, specially those concerning buildings and the relation to
Alperin’s solution. He also pointed out some flaws in the examples and in the
analysis of fixed points in Section 6 that are now fixed.
Organization of the paper: In Section 2 we introduce notation and basic facts
about the semidirects products we are interested in. In Section 3 we define, for each
element a ∈ A, a chain in top dimension r − 1 for |Ap (Frp )|. Then, in Section 4,
we add up these chains to form a chain in top dimension for |Ap (K ⋊ Frp )|. We

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ANTONIO DÍAZ RAMOS

also formulate the linear equations that must be satisfied in order for this chain to
c
e r−1
become a cycle in H
(|Ap (K ⋊ Frp )|; A). Case (a) of Theorem 1.4 and Theorem
1.5 are also proven. In Section 5, we introduce some tool needed to prove later in
the same section the abelian case Theorem 1.4(b). Afterwards, we prove Theorem
1.6. In the final Section 6, we prove Theorems 1.7, 1.8, 1.9 and 1.10.
2. Preliminaries
Throughout the paper, p denotes a prime and A denotes an abelian group. For a
poset P, its realization |P| is the topological realization of its order complex ∆(P),
i.e., of the simplicial complex whose n-simplices are the compositions p0 < p1 <
e ∗ (|P|; A) via the simplicial
. . . < pn in P. In particular, we compute the homology H
complex C∗ (∆(P); A). Here, Cn (∆(P); A) has one generator for each n-simplex of
∆(P ) and C−1 (∆(P); A) has as generator the empty composition
Pn ∅. The differential
d : Cn (∆(P); A) → Cn−1 (∆(P); A) is given as usual by d = i=0 (−1)i di , where di
forgets the i-th vertex of a given simplex. If G is a finite group and p is a prime,
let Ap (G) denote the Quillen poset of G at the prime p [8]. This poset consists
of the non-trivial elementary abelian p-subgroups of G ordered by inclusion. We
will pay special attention to Ap (G) for G a semidirect product G = K ⋊ H, where
H is an abelian p-group acting on the p′ -group K. Note that in this situation
Sylp (G) = K H = {kHk −1 , k ∈ K} and | Sylp (G)| = |K|/|CK (H)|. We introduce
the following notation.
Definition 2.1. Let G be the semidirect product G = K ⋊ H, where H is an
abelian p-group and K is a p′ -group. For any subgroup I ≤ G, we define N (I) as
the subset of Sylow p-subgroups of G that contain I and we also set N (I) = |N (I)|.
The next two results are basic coprime action properties about the semidirect
products we are concerned with.
Proposition 2.2. Let G be a semidirect product G = K ⋊ H, with H an abelian
p-group and K a p′ -group. Let I be a subgroup of H. Then:
(1) I is the only conjugate of I that lies in H.
(2) N (I) = |CK (I)|/|CK (H)|.
(3) If N E K is H-invariant then CK/N (H) = CK (H)N/N .
Proof. For the first claim, we reproduce here for convenience the proof in [7, Lemma
1.2]. So let g I be a conjugate of I that lies in H. Then we have g = kh for some
k ∈ K and h ∈ H and, as H is abelian, we have g I = k I. Now let i ∈ I. Then
i−1 (k i) ∈ H and i−1 (k i) = k i k −1 ∈ K. As K ∩ H = {1} we obtain i = k i,
k ∈ CK (I) and g I = k I = I. For the second claim, consider the action of CK (I) on
N (I) given by k · J = k J for k ∈ CK (I) and J ∈ N (I). This action is easily seen
to be transitive by the previous argument. Moreover, the isotropy group of J = H
is exactly CK (H). The second claim follows. The last item is standard and can be
found in [7, Lemma 2.2(c)].

From here on, we will only consider the case with H an elementary abelian pgroup of rank r, Frp . Then we will think of H as an Fp -vector space whose elements
are r-tuples of elements from Fp : (x1 , . . . , xr ) ∈ H with xi ∈ Fp . Consequently,
we call hyperplanes of H those subgroups I of H with |H : I| = p. We say
that a collection of hyperplanes is linearly independent if their corresponding onedimensional subspaces are independent in the dual vector space H ∗ .

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

5

Proposition 2.3. Let G be a semidirect product G = K ⋊ H, with H = Frp and
K a p′ -group. If H acts faithfully on K then there exists r linearly independent
hyperplanes of H, H1 , . . . , Hr , that satisfy N (Hi ) > 1.
Proof. A proof may be found in [7, Lemma 3.1] but we include here an alternative
argument: By [1, Exercise 8.1], we have K = hCK (I)|I hyperplane of Hi. Denote
by I the set of hyperplanes I of H that satisfy CK (H) ( CK (I), i.e., that, by
Proposition 2.2(2), satisfy N (I) > 1. The intersection ∩I∈I I acts trivially on K
because K is generated by {CK (I)}I∈I . Hence this intersection must be trivial as
the action is faithful. Now observe that the dimension of ∩I∈I I is exactly r minus
the maximal number of linearly independent hyperplanes in I.

For the case H = Frp , we also introduce the notation [i1 , . . . , il ] to denote a
sequence of elements from {1, . . . , r} with no repetition, and we say that two sequences are equal if they have the same length and the same elements in the same
order. Moreover, we denote by Slr the set of all such sequences which are of length
l. For a sequence [i1 , . . . , il ] ∈ Slr , we define the following subspace of H:
(6)

H[i1 ,...,il ] = {(x1 , . . . , xr ) ∈ H|xi1 = . . . = xil = 0}.

Thus H∅ = H, H[i] is the hyperplane with zero i-th coordinate and H[i1 ,...,il ] =
H[i1 ] ∩ . . . ∩ H[il ] ∼
= Fr−l
p . In addition, we also have
(7)

H[i1 ,...,il ] = H[j1 ,...,jl ] ⇔ {i1 , . . . , il } = {j1 , . . . , jl },

where on the right-hand side the equality is between (unordered) sets. We define
the signature ǫ[i1 ,...,il ] as
(8)

ǫ[i1 ,...,il ] = (−1)n+m ,

where n is the number of transpositions we need to apply to the sequence [i1 , . . . , il ]
to arrange it in increasing order [j1 , . . . , jl ], and m is the number of entries in which
[j1 , . . . , jl ] differ from [1, . . . , l]. It is immediate that the parity of n does not depend
on the particular choice of transpositions and so ǫ[i1 ,...,il ] is well defined. As an
example consider [1, 4, 3] ∈ S34 . Then we have ǫ[1,4,3] = (−1)1+2 = −1. We will
need the following property of the signature ǫ· :
r
Lemma 2.4. Let [i1 , . . . , ir−1 ] and [j1 , . . . , jr−1 ] be sequences in Sr−1
such that
[i1 , . . . , ir−2 ] = [j1 , . . . , jr−2 ] and ir−1 6= jr−1 . Then ǫ[i1 ,...,ir−1 ] + ǫ[j1 ,...,jr−1 ] = 0.

Proof. Write ǫ[i1 ,...,ir−1 ] = (−1)ni +mi and ǫ[j1 ,...,jr−1 ] = (−1)nj +mj as in (8). Without loss of generality we may assume that ir−1 < jr−1 and so we can order the set
{1, . . . , r} as follows:
(9)

1 < 2 < . . . < ir−1 < . . . < jr−1 < . . . < r.

Denote by n the number of transpositions needed to arrange [i1 , . . . , ir−2 ] in increasing order. Then, from (9), ni = n + r − 1 − ir−1 and nj = n + r − jr−1 . It
is also clear from (9) that mi = r − jr−1 and that mj = r − ir−1 . Hence we get
(nj + mj ) − (ni + mi ) = 1 and we are done.


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ANTONIO DÍAZ RAMOS

3. A chain for elementary abelian p-groups
Let H = Frp be an elementary abelian p-subgroup of rank r and let A be an
abelian group. We consider the poset Ap (H) of dimension r − 1 and its order complex ∆(Ap (H)). We shall explicitly define a chain in the abelian group
Cr−1 (∆(Ap (H)); A), i.e., a chain in top dimension r − 1. We start defining, for
[i1 , . . . , il ] ∈ Slr , the following l-simplex in ∆(Ap (H)):
(10)

σ[i1 ,...,il ] = H[i1 ,...,il ] < H[i1 ,...,il−1 ] < . . . < H[i1 ,i2 ] < H[i1 ] < H.

Remark 3.1. Note that, by repeated application of (7), we have that
σ[i1 ,...,il ] = σ[j1 ,...,ij ] ⇔ [i1 , . . . , il ] = [j1 , . . . , jl ].
Now, for any a ∈ A, we define the following chain in Cr−1 (∆(Ap (H)); A) :
X
ǫ[i1 ,...,ir−1 ] · a · σ[i1 ,...,ir−1 ] .
(11)
ZH,a =
r
[i1 ,...,ir−1 ]∈Sr−1

The next proposition exhibits the key feature of the chain ZH,a .
Proposition 3.2. The differential d(ZH,a ) ∈ Cr−2 (∆(Ap (H)); A) is given by
X
d(ZH,a ) = (−1)r−1
ǫ[i1 ,...,ir−1 ] · a · τ[i1 ,...,ir−1 ] ,
r
[i1 ,...,ir−1 ]∈Sr−1

where
τ[i1 ,...,ir−1 ] = H[i1 ,...,ir−1 ] < H[i1 ,...,ir−2 ] < . . . < H[i1 ,i2 ] < H[i1 ] .
Remark 3.3. Again by (7), we have τ[i1 ,...,il ] = τ[j1 ,...,ij ] ⇔ [i1 , . . . , il ] = [j1 , . . . , jl ].
Proof. The differential d : Cr−1 (∆(Ap (H)); A) → Cr−2 (∆(Ap (H)); A) is given by
Pr−1
the alternate sum i=0 (−1)i di . Note that, by the rank of the subspaces of H,
di (σ[i1 ,...,ir−1 ] ) = dj (σ[j1 ,...,jr−1 ] ) ⇒ i = j. We show below that di (ZH,a ) = 0 for
0 ≤ i ≤ r − 2. Then the theorem follows. We start considering d0 . Note that
d0 (σ[i1 ,...,ir−1 ] ) = H[i1 ,...,ir−2 ] < . . . < H[i1 ,i2 ] < H[i1 ] < H,
and that, by repeated application of (7), we have
d0 (σ[i1 ,...,ir−1 ] ) = d0 (σ[j1 ,...,jr−1 ] ) ⇔ [i1 , . . . , ir−2 ] = [j1 , . . . , jr−2 ].
Assume that such condition holds for the sequences [i1 , . . . , ir−1 ] and [j1 , . . . , jr−1 ].
Then they must be identical but for the last terms ir−1 and jr−1 . Moreover, if the
given sequences are different, we must have ir−1 6= jr−1 . Then, by Lemma 2.4,
ǫ[i1 ,...,ir−1 ] and ǫ[j1 ,...,jr−1 ] have opposite signs and the corresponding summands
cancel each other in the expression for d0 (ZH,a ). Hence d0 (ZH,a ) = 0.
Now we consider di for 0 < i ≤ r − 2. In this case we have:
di (σ[i1 ,...,ir−1 ] ) = H[i1 ,...,ir−1 ] < . . . < Ĥ[i1 ,...,ir−1−i ] < . . . < H[i1 ,i2 ] < H[i1 ] < H,
where the term H[i1 ,...,ir−1−i ] is missing. Again by (7), if the sequences [i1 , . . . , ir−1 ]
and [j1 , . . . , jr−1 ] have the same differential di , then they are identical but for
the entries ir−i−1 , ir−i and jr−i−1 , jr−i , and these entries satisfy {ir−i−1 , ir−i } =
{jr−i−1 , jr−i }. So, if the two given sequences are not the same one, then they differ
by a transposition. Hence ǫ[i1 ,...,ir−1 ] and ǫ[j1 ,...,jr−1 ] have opposite signs and their
corresponding summands cancel each other in the expression for di (ZH,a ). Thus
di (ZH,a ) = 0.


ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

7

Example 3.4. For r = 3 we have S23 = {[1, 2], [2, 1], [1, 3], [3, 1], [2, 3], [3, 2]} and
the signatures are +1, −1, −1, +1, +1 and −1 respectively. Within ∆(Ap (F3p )) we
have the following cover relations involving the six 2-simplexes σ[i1 ,i2 ] , [i1 , i2 ] ∈ S23 :
❡❡❡❡⑧ H[1] < H
❡❡❡❡❡❡ ⑧⑧⑧
❡
❡
❡
❡
❳
❳
H[1,2] < H[1] < H
❳❳❳❳❳ ⑧⑧♠♠♠ H[1,2] < H
❳❳♠❳♠⑧❳♠
♠♠⑧♠⑧⑧ ❳❳ H[1,2] < H[1]
♠
♠
♠
♠♠❳♠❳❳❳⑧⑧⑧
H[2,1] < H[2] < H ❳
✆ H[2] < H
❳
⑧⑧ ❳❳❳❳❳❳✆❳✆✆
⑧
✆ H[2,1] < H[2]
⑧⑧
✆✆ ♠♠ H[1,3] < H
⑧⑧❳❳❳❳
H[1,3] < H[1] < H ❳
✆
❳❳❳✆❳✆❳♠♠♠
♠ ❳❳❳ H
♠✆♠✆✆♠
[1,3] < H[1]
♠
♠
♠
♠♠❳♠❳❳✆❳✆✆
H[3,1] < H[3] < H ❳
H
②
[3] < H
✆ ❳❳❳❳❳❳❳②❳②②
✆
②
H
✆
[3,1] < H[3]
②②
✆❳✆
②
✆
❳
②
H[2,3] < H[2] < H
❳❳❳❳②❳②
♠♠♠ H[2,3] < H
②②♠♠❳♠❳♠❳♠❳♠❳❳ H
②
[2,3] < H[2]
② ♠
②♠②♠♠♠
②
♠
H
<H <H
H
<H .
[3,2]

[3]

[3,2]

[3]

The different signs for the signatures cause ZH,a
to have boundary d(ZH,a ) supported on the six
1-simplexes τ[i1 ,i2 ] with [i1 , i2 ] ∈ S23 . So the chain
ZH,a has support in the right-hand side coneshaped 2-dimensional simplicial complex.
Remark 3.5. The figure above suggest that ZH,a is a sort of cone construction.
In fact, it is straightforward that ZH,a is the join of the vertex H with the standard
apartment arising from the basis of the one-dimensional subspaces Fp vi , where the
vector vi = (0, . . . , 0, 1, 0, . . . , 0) has the 1 in the i-th position for i = 1, . . . , r.
4. A chain for semidirect products
Let H = Frp be an elementary abelian p-subgroup of rank r that acts on the
p -group K and consider the semidirect product G = K ⋊ H. We consider the
poset Ap (G) of dimension r − 1 and its order complex ∆(Ap (G)). We shall define a
chain Cr−1 (∆(Ap (G)); A) in top dimension r − 1 using the chain defined in Section
3. For each Sylow p-subgroup S ∈ Sylp (G) choose kS ∈ K with S = kS H and
choose an element aS of the abelian group A. Then consider the conjugated chain
′

ZS,aS := kS (ZH,aS ),

(12)

where ZH,aS ∈ Cr−1 (∆(Ap (H)); A) ⊆ Cr−1 (∆(Ap (G)); A) was defined in Equation
(11) and kS acts by conjugation on Cr−1 (∆(Ap (G)); A) and hence transforms ZH,aS
somewhere inside this chain space. The next lemma shows that ZS,aS does not
depend on the conjugating element kS :
Lemma 4.1. Let k1 , k2 ∈ K with
k2
(ZH,a ).

k1

H =

k2

H and let a ∈ A. Then

k1

Proof. By Equations (11) and (10), it is enough to see that
k1

H[i1 ,...,ir−1 ] < k1 H[i1 ,...,ir−2 ] < . . . < k1 H[i1 ,i2 ] < k1 H[i1 ] < k1 H

(ZH,a ) =

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ANTONIO DÍAZ RAMOS

and
k2

H[i1 ,...,ir−1 ] < k2 H[i1 ,...,ir−2 ] < . . . < k2 H[i1 ,i2 ] < k2 H[i1 ] < k2 H

are equal for each sequence [i1 , . . . , ir−1 ] of length r − 1. But note that for any
1 ≤ l ≤ r−1, the subgroups k1 H[i1 ,...,il ] and k2 H[i1 ,...,il ] are K-conjugates of H[i1 ,...,il ]
lying in k1 H = k2 H. Hence, by Proposition 2.2(1), these two subgroups must be
equal. The result follows.

Now, for a· = (aS )S∈Sylp (G) elements of A, and (kS )S∈Sylp (G) elements from K
satisfying kS H = S, we define the following element of Cr−1 (∆(Ap (G)); A):
X
X
kS
(ZH,aS ).
ZS,aS =
(13)
ZG,a· =
S∈Sylp (G)

S∈Sylp (G)

The lemma above shows that ZG,a· does not depend on the particular choice of the
elements kS ’s.
Proposition 4.2. The differential d(ZG,a· ) ∈ Cr−2 (∆(Ap (G)); A) is given by


X
X
ǫ[i1 ,...,ir−1 ] ·
aS ′ · kS τ[i1 ,...,ir−1 ] ,
d(ZG,a· ) = (−1)r−1
S ′ ∈N (kS H[i1 ] )

([i1 ,...,ir−1 ],S)∈T

r
× Sylp (G) and
where T is certain subset of Sr−1

τ[i1 ,...,ir−1 ] = kS H[i1 ,...,ir−1 ] < kS H[i1 ,...,ir−2 [ < . . . < kS H[i1 ,i2 ] < kS H[i1 ] .
P
kS
Proof. We have d(ZG,a· ) =
S∈Sylp (G) d( ZH,as ) and, by Proposition 3.2, the
kS

differential of

kS

ZH,as is given by

d(kS ZH,as ) = (−1)r−1

X

ǫ[i1 ,...,ir−1 ] · aS · kS τ[i1 ,...,ir−1 ] ,

r
[i1 ,...,ir−1 ]∈Sr−1

where τ[i1 ,...,ir−1 ] = H[i1 ,...,ir−1 ] < H[i1 ,...,ir−2 ] < . . . < H[i1 ,i2 ] < H[i1 ] .
r
× Sylp (G) the following equivalence
To study d(ZG,a· ), we define on the set Sr−1
relation:
([i1 , . . . , ir−1 ], S) ∼ ([j1 , . . . , jr−1 ], S ′ ) ⇔ kS τ[i1 ,...,ir−1 ] = kS′ τ[j1 ,...,jr−1 ] .
r
× Sylp (G) to be any set of representatives for the equivalence
We also set T ⊆ Sr−1
classes of ∼. Now we fix ([i1 , . . . , ir−1 ), S) ∈ T and study its ∼-equivalence class.
So assume that ([i1 , . . . , ir−1 ], S) ∼ ([j1 , . . . , jr−1 ], S ′ ). Then, in the first place,
we have kS H[i1 ] = kS′ H[j1 ] < kS′ H. In addition, for any 1 ≤ l ≤ r − 1, the
−1
subgroup kS′ kS H[i1 ,...,il ] = H[j1 ,...,jl ] is a subgroup of H that is K-conjugated to
H[i1 ,...,il ] ≤ H. Hence, by Proposition 2.2(1), we must have H[i1 ,...,il ] = H[j1 ,...,jl ] .
Using (7) we obtain then that [i1 , . . . , ir−1 ] = [j1 , . . . , jr−1 ]. So we have proven the
left to right implication of the next claim:

([i1 , . . . , ir−1 ], S) ∼ ([j1 , . . . , jr−1 ], S ′ ) ⇔ [i1 , . . . , ir−1 ] = [j1 , . . . , jr−1 ]
and

kS

H[i1 ] < kS′ H,

and the right to left implication is a direct consequence of Proposition 2.2(1) again.
It turns out then that the signature is constant along the ∼-equivalence class of
([i1 , . . . , ir−1 ], S) and the formula in the statement follows.


ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

9

Note that the chain ZG,a· is a cycle, i.e., d(ZG,a· ) = 0, if and only if
X
(14)
aS = 0
S∈N (k H[i] )

for all i ∈ {1, . . . , r} and all K-conjugates k H[i] of H[i] , and it is non-trivial if and
only if
(15)

aS 6= 0 for some S ∈ Sylp (G).

Definition 4.3. If equations (14) are satisfied we say that [ZG,a· ] is a constructible
e r−1 (|Ap (K ⋊ H)|; A).
class in H

c
e r−1
It is immediate that constructible classes form a subgroup H
(|Ap (K ⋊H)|; A)
e r−1 (|Ap (K ⋊H)|; A) as claimed in the introduction. For r = 1, i.e., for H = Fp ,
of H
and considering the “hyperplane” ∅ ∈ ∆(Ap (K ⋊ H)), Equations (14) and (15) just
e c (|Ap (K ⋊ H)|; A) = H
e 0 (|Ap (K ⋊ H)|; A).
say that H
0

Remark 4.4. If for some a ∈ A we choose aS = a for all S ∈ Sylp (G) then the
Equation for acyclicity of ZG,a· , (14), simplifies to:
(16)

N (H[i] ) · a = 0

for all i ∈ {1, . . . , r}.
Corollary 4.5. Let G be a semidirect product G = K ⋊ H with H = Frp and K
a p′ -group. Assume there are r linearly independent hyperplanes H1 , . . . , Hr such
c
e r−1
that D > 1 divides the numbers N (H1 ), . . . , N (Hr ). Then H
(|Ap (G)|; ZD ) 6= 0.

Proof. By a suitable change of basis in the Fp -vector space H, we may assume that
Hi = H[i] for i = 1, . . . , r. Set A = Z/DZ = ZD . Then, if we choose a = 1 ∈ A as
in Remark 4.4, we have N (Hi ) · 1 = 0 mod D for all i and it follows that ZG,a· is
c
e r−1
a non-trivial homology class in H
(|Ap (G)|; A).

Corollary 4.6. Let G be a semidirect product G = K ⋊ H with H = Frp acting
e c (|Ap (G)|; Zq ) 6= 0.
faithfully on the q-group K for a prime q 6= p. Then H
r−1

Proof. By Proposition 2.3, there exist r linearly independent hyperplanes H1 , . . . , Hr
that satisfy N (Hi ) > 1. Being K a q-group, this implies that 1 < q divides N (Hi )
c
e r−1
for all i. Then, by Corollary 4.5, H
(|Ap (G)|; Zq ) 6= 0.

This proves Theorem 1.4(a) of the introduction. Equations (14) form a system
Pr of homogeneous linear equations over the abelian group A. It consists of
i=1 |K|/|CK (H[i] )| equations over |K|/|CK (H)| variables and the entries of the
matrix associated to the system are either 0 or 1. Assume that A is a ring which is a
Principal Ideal Domain (P.I.D.). Then, if there are fewer equations than variables:
r
r
X
X
|K|
1
|K|
(17)
<
⇔
< 1,
|CK (H[i] )|
|CK (H)|
|N (H[i] )|
i=1
i=1
the Smith-Normal form of the matrix associated to the system shows that there
exists at least a non-trivial solution.
Corollary 4.7. Let G be a semidirect product G = K ⋊ H with H = Frp acting
faithfully on the p′ -group K with |K| = q1e1 · · · qlel . If r < qi for i = 1, . . . , l, then
c
e r−1
(|Ap (G)|; Z) 6= 0.
H

10

ANTONIO DÍAZ RAMOS

Proof. After a suitable reordering, we may assume that q1 < . . . < qr and hence
the hypothesis on r reads as r < q1 . As the integers Z form a P.I.D., we must
check that Equation (17) holds. By Proposition 2.3 and a suitable change of basis
in the Fp -vector space H we may assume that Hi = H[i] for i = 1, . . . , r with
|N (H[i] )| > 1. Because these numbers divide |K| we have |N (H[i] )| ≥ q1 and
Pr
r
1

i=1 |N (H[i] )| ≤ q1 < 1.
This result proves Theorem 1.5 of the introduction. In view of Equations (14),
it is worth considering the following graph.

Definition 4.8. For the semidirect product G = K ⋊ H with K a p′ -group and
H ∼
= Frp , let G be the graph with vertices all K-conjugates of H together with
all K-conjugates of the hyperplanes H[i] , and with edges all the inclusions among
them.
By Proposition 2.2(1), the vertices that are K-conjugates of H have degree
equal to r. A vertex which is K-conjugate to H[i] has degree N (H[i] ). Moreover,
Equations (14) can be interpreted as assigning the value aS with S ∈ Sylp (G) to
the vertex S ∈ Sylp (G) and, for each vertex k H[i] with k ∈ K, equalizing to zero
the sum of the values associated to its neighbours.
Example 4.9. Consider G = C3 × C3 ⋊ F2 × F2 with the generators acting by
(x1 , x2 ) 7→ (−x1 , x2 ) and (x1 , x2 ) 7→ (x1 , −x2 ) respectively. Set H1 = F2 × 0 and
H2 = 0 × F2 . Then we have CK (H1 ) = 0 × C3 , CK (H2 ) = C3 × 0, CK (H) = 1
and N (H1 ) = N (H2 ) = 3. The graph G is shown below, where upper-case letters
correspond to conjugates of H and lower-case letters to conjugates of hyperplanes.
By Corollary 4.6, there exists a non-trivial homology class ZG in the top dimension
homology group H1c (∆(A2 (G)); Z3 ). Moreover, by Corollary 4.7, as there are 6
e 1c (∆(A2 (G)); Z) 6= 0.
equations and 9 variables, we also have H
a

b

A B C D E F
d

e

c

G H

I

f

5. D-systems and quotients
Under some hypothesis, we have constructed above non-trivial classes in top
homology that are constant solutions (Remark 4.4) to the Equations (14). In this
section, we construct solutions which are characteristic functions. We first specialize
Definition 2.1.
Definition 5.1. Let G be a semidirect product G = K ⋊ H with H = Frp and K a
p′ -group. Let S ⊆ Sylp (G) be a non-empty subset of Sylow p-subgroups. For any
subgroup I ≤ G, we define NS (I) as the subset of Sylow p-subgroups of S that
contain I and we also set NS (I) = |NS (I)|. If D > 1 is an integer, we say that S
is a D-system if D divides NS (k H[i] ) for all i and all k ∈ K.

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

11

Theorem 5.2. Let G be a semidirect product G = K ⋊ H with H = Frp and K a
e c (|Ap (G)|; ZD ) 6= 0.
p′ -group. If S is a D-system then H
r−1
Proof. Form the chain ZG,a· as in Equation (13), where we choose aS = 1 for S ∈ S
and 0 otherwise. This is a not trivial chain and the Equations (14) for ZG,a· being
a cycle become
NS (k H[i] ) = 0 mod D,

for all i ∈ {1, . . . , r} and all K-conjugates k H[i] of H[i] . This holds by hypothesis.

Next we formulate a group theoretical condition that implies the existence of a
2-system.
Theorem 5.3. Let G be a semidirect product G = K ⋊ H with H = Frp and K
a p′ -group. Assume there are elements ci ∈ CK (H[i] ) \ CK (H) for i = 1, . . . , r
such [ci , cj ] = 1 for all i and j. Then there exists a 2-system for G and hence
e c (|Ap (G)|; Z2 ) 6= 0.
H
r−1
δ1 δ2

δr

Proof. Let S = {c1 c2 ...cr H, with δi ∈ {0, 1} } ⊆ Sylp (G). We claim that S is a
2-system: consider k H[i] for some i ∈ {1, . . . , r} and some k ∈ K. If NS (k H[i] ) = 0
then there is nothing to prove. Assume then that the set NS (k H[i] ) is not empty
δ1 δ2

δr

and choose one of its elements: c1 c2 ...cr H ∈ NS (k H[i] ) for some values δi ’s in
{0, 1}. Then by Proposition 2.2(1) we have
k

δ1 δ2
c2 ...cδrr

H[i] = c1

δ1 δ2
c2 ...cδrr

H[i] < c1

H.

Assume first that δi = 0. Because [ci , cj ] = 1 for all j, we have:
δ

δ

δ

i−1
i+1
ci+1
...cδrr
ci c11 ...ci−1

(

δi−1
δi+1
δ1
...ci−1
ci ci+1
...cδrr

H) = c1

δi−1 δi+1
δ1
...ci−1
ci+1 ...cδrr ci

H = c1

H.

δi−1 δi+1
δ
δ
ci c11 ...ci−1 ci+1 ...crr

From here we deduce that (
H[i] ) is a subgroup of the last two
subgroups in the display above. So, by Proposition 2.2(1) again, we must have:
δ

δ

δ

i−1
i+1
ci+1
...cδrr
ci c11 ...ci−1

(

δ1

H[i] ) = c1

δ

δ

i−1
i+1
...ci−1
ci ci+1
...cδrr

δ1

H[i] = c1

δ

δ

i−1
i+1
...ci−1
ci+1
...cδrr ci

H[i] .

As ci centralizes H[i] , the right-hand side group in the equation above is equal to
δ

δ

δ

i−1
i+1
c11 ...ci−1
ci+1
...cδrr

H[i] . Hence,

δ

δ

δ

i−1
i+1
δ
ci c11 ...ci−1 ci+1 ...crr

(

H) ∈ NS (k H[i] ). Moreover, this

δi−1 δi+1
δ
ci+1 ...cδrr
c11 ...ci−1

H because then we would
K-conjugate of H cannot be equal to
obtain ci ∈ CK (H), a contradiction. So, we have shown, in the case δi = 0, that
δ

δ

c11 ...ci i ...cδrr

δ1

H ∈ NS (k H[i] ) ⇒ c1

1−δi

...ci

...cδrr

H ∈ NS (k H[i] ),

and both K-conjugates of H are different. The case δi = 1 is proven analogously
using conjugation by c−1
instead of by ci . It follows that NS (k H[i] ) is an even
i
number.

Remark 5.4. The set S defined in the proof of Theorem 5.3 has size 2r : Consider
non-trivial elements hi ∈ ∩j6=i H[j] . Then H = hh1 , . . . , hr i. Moreover, if we have
ǫi
ǫ
c11 ...cǫrr
H = H with ǫi ∈ {−1, 0, 1}, we obtain, by Proposition 2.2(1), that ci hi = hi
for all i. If there exists i0 with ǫi0 6= 0, we deduce that [ci0 , hi0 ] = 1 and then, as
H = hH[i0 ] , hi0 i, that ci0 ∈ CK (H), a contradiction.

12

ANTONIO DÍAZ RAMOS
P

δ1 δ2

δr

Consider then the chain ZG,b· , where bS = (−1) i δi if S = c1 c2 ...cr H ∈ S
c
e r−1
(|Ap (G)|; Z)
and bS = 0 otherwise. It represents a homology class [ZG,b· ] ∈ H
c
e
is
exactly
the cycle
(|A
(G)|;
Z
).
In
fact,
Z
which is a lift of [ZG,a· ] ∈ H
p
2
G,b·
r−1
corresponding to the join of the r 0-spheres {h1 , c1 h1 }, . . . , {hr , cr hr }.
Corollary 5.5. Let G be a semidirect product G = K ⋊ H with H = Frp acting
faithfully on the abelian p′ -group K. Then there exists a 2-system for G and hence
c
Hr−1
(|Ap (G)|; Z2 ) 6= 0.
Proof. By Proposition 2.3, there exist r linearly independent hyperplanes H1 , . . . , Hr
that satisfy N (Hi ) = |CK (Hi )|/|CK (H)| > 1. By a suitable change of basis in the
Fp -vector space H, we may assume that Hi = H[i] for i = 1, . . . , r. As K is abelian,
any choice of elements ci ∈ CK (H[i] ) \ CK (H) 6= ∅ for i = 1, . . . , r satisfies the
hypothesis of Theorem 5.3.

This prove Theorem 1.4(b). We finish this section by proving Theorem 1.6 of
the introduction.
Theorem 5.6. Let G be a semidirect product G = K ⋊ H with H = Frp and K
a p′ -group and let N E K be a normal H-invariant subgroup of K. Consider the
quotient Ḡ = G/N and let A be an abelian group. Then every class [ZḠ,ā· ] ∈
c
e c (|Ap (G)|; A). Moreover, a
Hr−1
(|Ap (Ḡ)|; A) can be lifted to a class [ZG,a ] ∈ H
r−1
non-trivial class lifts to a non-trivial class.
Proof. Consider the projection homomorphism π : G → Ḡ. Because its kernel is
the p′ -order subgroup N , there is an induced surjective application:
Sylp (π) : Sylp (G) → Sylp (Ḡ).
Moreover, for π(S) ∈ Sylp (Ḡ), we have Sylp (π)−1 (π(S)) = {n S, n ∈ N } by Proposition 2.2(3). We have the collection of elements of A, ā· = (āS̄ )S̄∈Sylp (Ḡ) , and we
define a· = (aS )S∈Sylp (G) by aS = āπ(S) . Now choose a K-conjugate k H[i] of H[i] .
Then the equality
Sylp (π)(N (k H[i] )) = N (π(k) π(H[i] ))
follows from Proposition 2.2(3). Equation (14) for ZG,a at k H[i] become:
X
X
X
aS =
āπ(S) = 0.
S∈N (k H[i] )

π(S)∈N (π(k) π(H[i] ))

S ′ ∈N (k H[i] )∩Syl−1
p (π)(π(S))


n
k
Note that N (k H[i] ) ∩ Syl−1
p (π)(π(S)) = { S, n ∈ N ∩ CK ( H[i] )} and that this
k
set has order m = |CN ( H[i] )|/|CN (S)|. As N is normal in K, the denominator
|CN (S)| does not depend on the Sylow p-subgroup S ∈ N (k H[i] ) and it is clear that
m does not depend neither. Hence, we may rewrite the equation above as
X
āπ(S) = 0,
m·
π(S)∈N (p(k) π(H[i] ))

and this equation holds because ZḠ,ā is a cycle. It is immediate from the construction that non-trivial classes lifts to non-trivial classes.

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

13

Example 5.7. Consider G = C5 × C5 × C5 ⋊ F2 × F2 × F2 with the generators
acting by sending (x1 , x2 , x3 ) to (−x1 , x2 , x3 ), (x1 , −x2 , x3 ) and (x1 , x2 , −x3 ) respectively. Set H1 = 0 × F2 × F2 , H2 = F2 × 0 × F2 and H3 = F2 × F2 × 0. Then
we have CK (H1 ) = C5 × 0 × 0, CK (H2 ) = 0 × C5 × 0, CK (H3 ) = 0 × 0 × C5 ,
N (Hi ) = 25 and CK (H) = 1. Although Corollaries 4.6 and 4.7 apply in this case,
we focus on the construction in Corollary 5.5. Set c1 = (1, 0, 0), c2 = (0, 1, 0) and
c3 = (0, 0, 1). Then the elements ci ∈ CK (Hi ) \ CK (H) satisfy the hypothesis
of Theorem 5.3. The 2-system constructed in the proof of this result is exactly
S = {H, c1 H, c2 H, c3 H, c1 c2 H, c1 c3 H, c2 c3 H, c1 c2 c3 H} and has size 8. The subgraph
of the graph G (Definition 4.8) containing S and its neighbours is the following.
c1

c2

c3

c1 c2

c1

c2

c2

c1 c3

c2 c3

c1 c2 c3

c3

c1 c2

H
❊
❊❘
❳H
❘
❙
❳H ❳ ❯
❲❙❲❙❲❙❲❤❲❤❤①❳H
❑H
❤H
✐H
❥H
❳❳❳❳❑❯❳❑❯❑❯❑❯❯❯❯
❲①❙❣❲❣❲❳❣❲❳❣❲❳❣❳❣❳❣❳❣❳❳❳❦❳❳❦❳❦❳❦❳❦❳❦❳❳❳❳❳❳
❤❤❙
q✂q❥✂q✂❥❥❥❥❥✐✐✐❊✐❊❊✐✐✐❤❤❤❊❤❊❘❤❊❘❤❘❤
❤
❙
①
❘
❤
q
❤
❘
❳
❣
❤
①
❳
q
❙
❦
❳
❲
❊
❊
❘
❣
❑❳ ❯❯
❤
❤ ❣❘❣❘①❣①
❳❳
❳❳ ❳❳❳
❙ ❲❲ ❦
q❥q❥q❥✂❥✂✐✂❥✐✐✐✐❤✐❤✐❤❤❤❤❤❤❤❤❊❊❤❊❤❤❤❣❤❣❊❣❊❣❊❣①
① ❘❘❘❘ ❦❙❦❙❦❙❦❙❦❙❦❙❲❲❲❲❲❲❲❳❲❳❳❳❳❳❳❳❳❳❳❳❳❳❳❑❳❑❳❳❑❳❳❑❳❳❯❳❯❳❯❳❯❳❯❳❯❳❯
q
❥
q
❥
✐
❣
❤
❤
❣
❦
❤
①
✂
✐
❤
❥
q

H1

H2

H3

c1

H2

H3

H1

H3

c3

H1

H2

H3

c1 c3

H2

c2 c3

H1

e 2c (|Ap (G)|; Z2 ) has
The homology class [ZG,a· ] ∈ H
support in the 2-dimensional simplicial complex
on the right-hand side. The eight faces of the
hollow octahedron are in bijection with the set
S. For each element of S, its corresponding face
is decomposed as in Example 3.4, that is, via its
barycentric subdivision (shown in one of the faces
of the figure).
6. Quillen’s conjecture
In this section we prove the results of the introduction directly related to Quillen’s
conjecture. We start with Theorem 1.7.
Theorem 6.1. QDpc,Z2 holds for K ⋊ H with H = Frp and K a solvable p′ -group.
Proof. As Op (K ⋊ H) = 1 we have that the action of H on K is faithful. Because
K ⋊ H is solvable, its Fitting group F (K ⋊ H) = F (K) is self-centralizing [1, 31.10]
and hence H also acts faithfully on the nilpotent group F (K). Now, the Frattini
quotient F (K)/Φ(F (K)) is abelian and, by a result of Burnside [4, 5.1.4], H acts
e c (|Ap (F (K)/Φ(F (K)) ⋊ H)|; Z2 ) 6=
faithfully on it. Next, by Corollary 5.5, H
r−1
e c (|Ap (F (K) ⋊ H)|; Z2 ) 6= 0. Finally,
0. Then by Theorem 5.6 we also have H
r−1
e c (|Ap (F (K) ⋊ H)|; Z2 ) ≤ H
e c (|Ap (K ⋊ H)|; Z2 ) finishes the
the inclusion H
r−1
r−1
proof.

Next we prove Theorem 1.8 of the introduction.
Theorem 6.2. Quillen’s conjecture holds for solvable groups.
Proof. We reproduce Quillen’s argument that reduces this result to the configuration of Theorem 6.1: Let G be a solvable group with Op (G) = 1. Let r be the
p-rank of G and choose H ∈ Ap (G) of rank r. By Hall-Highman [6, Lemma 1.2.3]
we have that K = Op′ (G) is self centralizing and hence H acts faithfully on this
c
e r−1
solvable group K. By Theorem 6.1 we have that H
(|Ap (K ⋊ H)|; Z2 ) 6= 0. The
e
e
inclusion Hr−1 (|Ap (K ⋊ H)|; Z2 ) ≤ Hr−1 (|Ap (G)|; Z2 ) finishes the proof.


14

ANTONIO DÍAZ RAMOS

Next, we focus on the p-solvable case. We start describing fixed points for
actions on direct products. We say that the group H acts by permutations on the
direct product Y = X1 × . . . × Xm if H acts on Y and each h ∈ H permutes the
groups {X1 , . . . , Xm } among themselves. This is the case, for instance, if each Xi
is quasisimple, by the Krull-Schmidt Theorem [5, 3.22].
Lemma 6.3. Let H be an elementary abelian p-group acting by permutations on the
m-fold direct product Y = X ×. . .×X and transitively permuting these components.
Then the following hold:
(1) If H ′ = NH (X) and H ′′ satisfies H = H ′ × H ′′ then
Y
h
CY (H) = {
x with x ∈ CX (H ′ )}.
h∈H ′′

(2) For I ≤ H, H ′ = NH (X), I ′ = NI (X), and k equal to the number of orbits
for the permutation action of I on the components of Y , we have:
h
i
h
i
′
′
′
CY (I) > CY (H) ⇔ k = 1 and CX (I ) > CX (H ) or k > 1 and CX (I ) > 1 .
Proof. Set H ′ = NH (X) and choose H ′′ ≤ H such that H = H ′ × H ′′ . Then we
have Y = h1 X × h2 X × . . . × hm X, where H ′′ = {h1 = 1, h2 , . . . , hm }. Moreover,
CY (H) = CY (H ′ ) ∩ CY (H ′′ ) and we
Q centralizers separately.
Q compute these two
For the former, we have CY (H ′ ) = h∈H ′′ Ch X (H ′ ) = h∈H ′′ h CX (H ′ ). For the
′′
latter, notice that
Q H hregularly permutes the components of Y . The map X′′ → Y
given by x 7→ h∈H ′′ x is a homomorphism which image is exactly CY (H ) (cf.
[5, Lemma 3.27]). The expression in (1) follows.
Regarding (2), write Y = Y1 · · · Yk , where each Yl is an orbit for the permutation
action of I on the components of Y . Then CY (I) = CY1 (I) · · · CYk (I). For the
action of I on each orbit Yl and with the notation of (1), we have I ′ = I ∩ H ′ and
we may choose I ′′ and H ′′ such that I ′′ = I ∩ H ′′ . Moreover, if J ≤ H ′′ satisfies
H ′′ = I ′′ × J, then we may write J = {j1 , . . . , jk } with Yl the I-orbit of jl X:
Y
i jl
Yl =
( X).
i∈I ′′

′

|
Note that k = |J| = |H ′′ : I ′′ | = |H:H
|I:I ′ | . The result follows from inspection of the
expressions in (1) for the centralizers CY (H) and CYl (I) for l = 1, . . . k.


The next result is of independent interest and it is the only place where we use
the Classification of the Finite Simple Groups (CFSG).
Proposition 6.4. Let H be an elementary abelian p-group acting on a direct product Y of m copies of a nonabelian simple p′ -group X and transitively permuting
these components. Let H1 , . . . , Hn be linearly independent hyperplanes of H such
that CY (Hi ) > CY (H) for all i. Then there exist elements ci ∈ CY (Hi ) \ CY (H)
such that [ci , cj ] = 1 for all i, j.
Proof. Write H ′ = NH (X), Hi′ = NHi (X) and choose H ′′ and Hi′′ for each i such
that H = H ′ × H ′′ and Hi = Hi′ × Hi′′ . As Hi′ ≤ H ′ and |H : Hi | = p, we have:
(1) either H ′ = Hi′ and |H ′′ |/|Hi′′ | = p,
(2) or |H ′ : Hi′ | = p and |H ′′ | = |Hi′′ |.

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

15

In case (1), we have CX (H ′ ) = CX (Hi′ ) and, by Lemma 6.3(2), the permutation
action of Hi on Y has exactly p orbits and |CX (Hi′ )| > 1. In case (2), Lemma
6.3(2) implies that CX (Hi′ ) > CX (H ′ ). If CH (X) = NH (X), then also CHi (X) =
NHi (X) and CX (H ′ ) = CX (Hi′ ) = X, a contradiction. Otherwise, AutH (X) =
NH (X)/CH (X) is a non-trivial group of outer automorphisms of X. But:
(18) If P is a p-group and 1 6= P ≤ Out(X) for a nonabelian simple p′ -group X,
then X is of Lie type and P is cyclic and consists of field automorphisms.
The proof of the claim (18) is an exhaustive check via the CFSG. Details are provided in [9, Theorem 8.2.12(2)] and the same argument is used in [2]. Hence,
AutH (X) is cyclic of order p. Then, either |CH (X) : CHi (X)| = |NHi (X) :
CHi (X)| = p or these two indexes are equal to 1. In the former case, AutHi (X)
is a non-trivial subgroup of AutH (X) and hence AutHi (X) = AutH (X). Then
H ′ = Hi′ + CH (X) and CX (H ′ ) = CX (Hi′ ), contradiction again. So the latter case
must hold, i.e., CH (X) = CHi (X) = NHi (X). We deduce that CX (Hi′ ) = X and,
from (18), that CX (H ′ ) = X σ for a field automorphism σ of X of order p. To sum
up, we conclude that for each hyperplane Hi :
(1) either Hi has p permutation orbits on Y and CX (Hi′ ) = CX (H ′ ) > 1,
(2) or Hi transitively permutes Y and CX (Hi′ ) = X > CX (H ′ ) = X σ .
Now we construct the elements ci in the statement.QIf all hyperplanes are of type
(1) then pick c ∈ CX (H ′ ) with c 6= 1 and set ci = h∈H ′′ h c for all i. Otherwise,
i
some hyperplane is of type (2), X is of Lie type and we notice that:
There exists c ∈ X σ and d ∈ X \ X σ such that [c, d] = 1.
The elements c and d may be chosen as elements in the maximal
torus of X satQ
isfying σ(c) = c and σ(d) 6= d. For Hi of type (1), set ci = h∈H ′′ h c. For Hi of
i
Q

type (2), set ci = h∈H ′′ h d. These elements satisfy [ci , cj ] = 1 for all i, j.

Remark 6.5. Under the hypotheses of Proposition 6.4, it also holds that n ≤ m.
Theorem 6.6. QDpc,Z2 holds for K ⋊ Frp with K a p′ -group.

Proof. As Op (K ⋊ H) = 1 we have that the action of H on K is faithful. Now, the
generalized Fitting subgroup F ∗ (K ⋊ H) = F ∗ (K) is self-centralizing [1, 31.13] and
hence H acts faithfully on it. Here, F ∗ (K) = F (K)E(K) with [E(K), F (K)] = 1
and where E(K) is the layer, which is a central product of quasisimple groups.
Moreover, By [5, 3.15(v)], Φ(F ∗ (K)) = Φ(F (K))Z(E(K)) and by [5, 3.5(v)],
F (K) ∩ E(K) = Z(E(K)). Then by [5, 3.9(ii)], it follows that F ∗ (K)/Φ(F ∗ (K)) =
A × B, where A is elementary abelian and B is a direct product of nonabelian
simple p′ -groups. By Burnside [4, 5.1.4], H acts faithfully on D = A × B, and we
are going to show that the hypothesis of Theorem 5.3 holds for D ⋊ H. Then the
same argumentsQas in Theorem 6.1 finish the proof.
Write B = j Bj , where each Bj is an H-orbit for the permutation action
of H on Q
the direct factors of B. For any subgroup I ≤ H we have CD (I) =
CA (I) × j CBj (I). By Proposition 2.3, there are r linearly independent hyperplanes of H, H1 , . . . , Hr such that |CD (Hi )|/|CD (H)| > 1. Hence, for each i,
either |CA (Hi )|/|CA (H)| > 1 or |CBj (Hi )|/|CBj (H)| > 1 for some j. In the former case, let c′i ∈ CA (Hi ) \ CA (H) by any element in this non-empty set and set

16

ANTONIO DÍAZ RAMOS

ci = (c′i , 1) ∈ CD (Hi ) \ CD (H). To choose ci for the latter case, consider, for a fixed
orbit Bj , the following set:
Bj = {i ∈ {1, . . . , r} such that |CBj (Hi )|/|CBj (H)| > 1}.
Let Bj = X × r2 X × . . . rm X, where X is a simple nonabelian p′ -group and r1 =
1, r2 , . . . , rm are representatives for the set H/NH (X) of size m. By Proposition 6.4,
there are elements c′′i ∈ CBj (Hi ) \ CBj (H) for each i ∈ Bj such that [c′′i , c′′i′ ] = 1 for
all i, i′ ∈ Bj . Then define c′i = (1, 1, . . . , 1, c′′i , 1, . . . , 1) ∈ CB (Hi ) \ CB (H), where
we place c′i in the position of the orbit Bj . Define ci = (1, c′i ) ∈ CD (Hi ) \ CD (H). It
is straightforward that the chosen elements ci ’s satisfy the hypothesis of Theorem
5.3, i.e., they commute pairwise.

r1

We finally prove Theorem 1.10
Theorem 6.7. Quillen’s conjecture holds for p-solvable groups.
Proof. As Hall-Highman [6, Lemma 1.2.3] is valid for p-soluble groups, the argument employed in the proof of Theorem 6.2 can be used here but for replacing
Theorem 6.1 by Theorem 6.6.

Final Remark 6.8. Here we give more details on Alperin’s approach compared
to the approach on this work. The reductions for the former route are as follows,
where K is a p′ -group:
G p-solvable
G solvable

/o /o o/(2)/o /o /o /o // K ⋊ Fr
p
O
CFSG

/o (1)
/o /o // K ⋊ Frp , K solvable

Alperin

/o o/ o/ // Join of r 0-spheres

Here, CFSG refers to the argument (18) already used in the proof of Proposition 6.4.
The rightmost reduction on the bottom line consists of Alperin’s coprime action
arguments leading to a minimal counterexample for which the join of spheres may
be considered. Our approach can be summarized as follows:
G p-solvable

G solvable

/o /o (2)
/o /o o/ o/ // K ⋊ Frp , K = A × S
o/ /o /o /o o/ // K ⋊ Frp /o /o /o 5.6
*j 5.3+CFSG
*j *j *j
*j **
4t 44 5.2
t
4
4t 4t
4t 5.3+abelian
t
4
/o (1)
/o /o /o // K ⋊ Fr , K = A
/o /o // K ⋊ Frp , K solvable /o 5.6
p

Here, A is an abelian p′ -group and S is a p′ -group that is a direct product of
nonabelian simple groups. Theorem 5.6 lifts classes from the quotient, and it is a
tool that was not available before. Theorem 5.3 implies the existence of a 2-system,
and it is a generalization to possibly non-solvable groups of the join of spheres
construction (see Remark 5.4).
References
[1] M. Aschbacher, Finite group theory, Second edition. Cambridge Studies in Advanced Mathematics, 10. Cambridge University Press, Cambridge, 2000.
[2] M. Aschbacher, S. Smith, On Quillen’s conjecture for the p-groups complex, Ann. of Math.
(2) 137 (1993), no. 3, 473-529.
[3] J.L. Alperin, A Lie approach to finite groups, GroupsCanberra 1989, 19, Lecture Notes in
Math., 1456, Springer, Berlin, 1990.

ON QUILLEN’S CONJECTURE FOR p-SOLVABLE GROUPS

17

[4] D. Gorenstein, Finite groups, Second edition. Chelsea Publishing Co., New York, 1980.
[5] D. Gorenstein, R. Lyons, R. Solomon, The classification of the finite simple groups. Number
2. Part I. Chapter G, General group theory. Mathematical Surveys and Monographs, 40.2.
American Mathematical Society, Providence, RI, 1996.
[6] P. Hall, G. Higman, On the p-length of p-soluble groups and reduction theorems for Burnside’s
problem, Proc. London Math. Soc. (3) 6 (1956), 1-42.
[7] T. Hawkes, I.M. Isaacs, On the poset of p-subgroups of a p-solvable group, J. London Math.
Soc. (2) 38 (1988), no. 1, 77-86.
[8] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. Math.
28 (1978), no. 2, 101-128.
[9] S.D. Smith, Subgroup complexes, Mathematical Surveys and Monographs, 179. American
Mathematical Society, Providence, RI, 2011.
Departamento de Álgebra, Geometrı́a y Topologı́a, Universidad de Málaga, Apdo
correos 59, 29080 Málaga, Spain.
E-mail address: adiazramos@uma.es

