ON A STRONG FORM OF OLIVER’S p-GROUP CONJECTURE

DAVID J. GREEN, LÁSZLÓ HÉTHELYI, AND NADIA MAZZA

Abstract. We introduce a stronger and more tractable form of Oliver’s pgroup conjecture, and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group S_{n} and of the general linear group GL_{n}(𝔽_{q}) satisfy both the strong conjecture and its reformulation.

1. Introduction

Bob Oliver proved the Martino–Priddy Conjecture, which states that the p-local fusion system of a finite group G uniquely determines the p-completion of its classifying space BG [10, 11]. Finite groups are not however the only source of fusion systems: the p-blocks of modular representation theory have fusion systems too, and there are other exotic examples as well. An open question in the theory of p-local finite groups claims that every fusion system has a unique p-completed classifying space – see [1] for a survey article on this field.

In [10], Oliver introduced the characteristic subgroup 𝔛(S) of a finite p-group S. For odd primes he showed that unique existence of the classifying space would follow from the following conjecture. Recall that J(S) denotes the Thompson subgroup of S generated by all elementary abelian subgroups of greatest rank.

Conjecture 1.1 (Oliver; Conjecture 3.9 of [10]). Let p be an odd prime and S a finite p-group. Then J(S) ≤ 𝔛(S).

In [4], the first two authors and Lilienthal obtained the following reformulation of Oliver’s conjecture.

Conjecture 1.2 (Conjecture 1.3 of [4]). Let p be an odd prime and G a finite p-group. If the faithful 𝔽_{p}G module V is an F -module, then there is an element 1 ≠ g ∈ Ω_{1}(Z(G)) such that the minimal polynomial of the action of g on V divides (X − 1)^{p−1}.

Recall that V is by definition an F -module if there is an offender, i.e., an elementary abelian subgroup 1 ≠ E ≤ G such that dim(V)−dim(V^{E}) ≤ rank(E). Theorem 1.2 of [4] states that Conjecture 1.1 holds for every p-group S with S/𝔛(S) ≅ G if and only if Conjecture 1.2 holds for G. Note that by [4, Lemma 2.3], every p-group G does occur as some S/𝔛(S).

Date: 25 May 2010.

2000 Mathematics Subject Classification. Primary 20D15.

Key words and phrases. p-group; offending subgroup; quadratic offender; p-local finite group.

Héthelyi supported by the Hungarian Scientific Research Fund OTKA, grant 77-476.

In this paper we modify Oliver’s construction of 𝔛(S) slightly to obtain a new characteristic subgroup Y(S), with Y(S) ≤ 𝔛(S) for p odd and Y(S) = S for p = 2: see §2. We therefore propose the following strengthening of Oliver’s conjecture:

Conjecture 1.3. Let p be a prime and S a finite p-group. Then J(S) ≤ Y(S).

This is indeed a strengthening: if p = 2 then Y(S) = S and so Conjecture 1.3 is true, whereas Oliver’s conjecture is false in general; and for p ≥ 3 we have Y(S) ≤ 𝔛(S). We recently learnt that Justin Lynd raises the same question in his paper [8]; note that his 𝔛_{3}(S) is our Y(S).

This conjecture admits a module-theoretic reformulation as well. An element g ∈ G is said to be quadratic on the 𝔽_{p}G-module V if its action has minimal polynomial (X − 1)^{2}. Note that if V is faithful then quadratic elements must have order p.

Conjecture 1.4. Let p be a prime and G a finite p-group. If the faithful 𝔽_{p}Gmodule V is an F -module, then there are quadratic elements in Ω_{1}(Z(G)).

Observe that Conjecture 1.4 is a strengthening of Conjecture 1.2, and trivially true for p = 2. Our first actual result is the equivalence of the two new conjectures:

Theorem 1.5. Let p be an odd prime and G a finite p-group. Then Conjecture 1.3 holds for every finite p-group S with S/Y(S) ≅ G if and only if Conjecture 1.4 holds for every faithful 𝔽_{p}G-module V.

Proof. The proof of [4, Theorem 1.2] adapts easily to the present case, given the results on Y -series and on Y(G) in Lemma 2.1. □

Note that Conjecture 1.4 holds if G is metabelian or of (nilpotence) class at most four: for though [5, Theorem 1.2] only purports to verify the weaker Conjecture 1.2, the proof that is given there only uses the assumptions of Conjecture 1.4.

Our main result is the following:

Theorem 1.6. For n ≥ 1 and a prime p, let P be a Sylow p-subgroup of the symmetric group S_{n}. Then Conjecture 1.3 holds with S = P, and Conjecture 1.4 holds with G = P.

Proof. For p = 2 there is nothing to prove, so assume that p ≥ 3. It is well known (see [6, III.15]) that P is a direct product P = ∏_{i=1}^{m} P_{r_{i}}, where r_{i} ≥ 1 for all i, and P_{r} is the iterated wreath product C_{p} ≀ C_{p} ≀ ⋯ ≀ C_{p} with r factors C_{p}. Recall that the Sylow p-subgroups of S_{p^{r}} are isomorphic to P_{r}.

As P_{1} ≅ C_{p} is abelian, and P_{r} satisfies Hypothesis 5.2 for r ≥ 2 by Corollary 6.6, Lemma 5.3 tells us that G = P satisfies Conjecture 1.4.

For Conjecture 1.3, observe that J(G_{1} × G_{2}) = J(G_{1}) × J(G_{2}) for any groups G_{1}, G_{2}. Huppert shows in [6, Satz III.15.4a)] that J(P_{r}) is an abelian normal subgroup of P_{r}. Hence J(P) is abelian and normal in P. So J(P) ≤ Y(P) by Lemma 2.1 (3). □

The fact that they provide the proper degree of generality for the proof of [5, Theorem 1.2] was not the only reason for introducing Y(S) and its companion Conjecture 1.4.

Theorem 1.7. Let p be a prime and G a finite p-group. Suppose that G satisfies one of the following conditions:

(1) G is generated by its abelian normal subgroups; or more generally

(2) Y(G) = G; or more generally

(3) Ω_{1}(Z(Y(G))) = Ω_{1}(Z(G)). Then Conjecture 1.4 holds for G.

Proof. Clearly (2) ⇒ (3). For (1) ⇒ (2), note from Lemma 2.1 (3) that every abelian normal subgroup lies in Y(G).

Now suppose that V is a faithful F -module. Timmesfeld’s replacement theorem in the version given in [5, Theorem 4.1] states that there are quadratic elements in G. Hence by Lemma 8.1 there are quadratic elements which are late in the sense of §8. But Lemma 8.2 says that if a quadratic element is late then it lies in Ω_{1}(Z(Y(G))). So if (3) holds then there are quadratic elements in Ω_{1}(Z(G)). □

For p ≥ 5, this result has been obtained independently by Justin Lynd: see the remark after [8, Lemma 8]. One application of this result is the following:

Theorem 1.8. Let n ≥ 1, let p be a prime, and let q be any prime power. Let P be a Sylow p-subgroup of the general linear group GL_{n}(𝔽_{q}). Then Conjecture 1.3 holds for S = P, and Conjecture 1.4 holds for G = P.

Proof. First we consider the case where q is a power of p (defining characteristic). If we can show that P is generated by its abelian normal subgroups then we are done: for this is condition (1) of Theorem 1.7, which implies both condition (2) and the conclusion of that theorem. Now, we may choose P to be the group of upper triangular matrices with ones on the diagonal. This copy of P is generated by the collection of abelian normal subgroups N_{i,j} for 1 ≤ i < j ≤ n, where

N_{i,j} = {A ∈ P | A_{aa} = 1; A_{ab} = 0 for a ≠ b whenever a > i or b < j} .

See also [13].

Now we turn to the case where (p, q) = 1 (coprime characteristic). It was first shown by Weir [12] that P is a direct product of iterated wreath products C_{p^{r}} ≀ C_{p} ≀ ⋯ ≀ C_{p}, where the C_{p^{r}} is the “innermost” factor in the wreath product. These iterated wreath products satisfy Hypothesis 5.2 by Lemma 5.6 (1) and Proposition 6.1. The one exception is of course C_{p^{r}}, which is abelian. So any direct product of these groups satisfies Conjecture 1.4 by Lemma 5.3.

We now turn to Conjecture 1.3 in coprime characteristic. As Y(P) = P if p = 2, we may assume that p is odd. We shall show that J(P) is abelian, from which J(P) ≤ Y(P) follows by Lemma 2.1 (3). Since J(G × H) = J(G) × J(H), we may assume that P is one iterated wreath product. Applying Lemma 5.4, we see that J(P) is elementary abelian by induction on the number of iterations. □

The method of Theorem 1.7 can also be used to show the following:

Theorem 1.9. Let p be a prime, G a finite p-group and V a faithful 𝔽_{p}G-module with no quadratic elements in Ω_{1}(Z(G)). Then the subgroup generated by all quadratic elements of G is a proper subgroup of G.

Proof. Let H ≤ G be the subgroup generated by all last quadratic elements, c.f. §8. By Lemma 8.1 we know that H ≠ 1. So H ≰ Z(G), for Z(G) has no quadratic elements. Hence C_{G}(H) ⪇ G. But C_{G}(H) contains every quadratic element by Lemma 8.5. □

Structure of the paper. In Section 2 we construct and study the characteristic subgroup Y(G) of G. We then prove some orthogonality relations in §3 and study weakly closed elementary abelian subgroups in §4. Then in Section 5 we introduce a hypothesis which constitutes a strengthening of Conjecture 1.4 for groups with cyclic centre. In Section 6 we demonstrate the key property of this hypothesis: it remains valid when passing from a group P to the wreath product P ≀ C_{p} = P^{p} ⋊ C_{p}.

In Section 7 we introduce the notions of deepest commutators and locally deepest commutators. In Section 8 we use late and last quadratics to demonstrate some lemmas which are required by the proofs of Theorem 1.7 and 1.9. Finally we show in §9 that powerful p-groups satisfy Conjecture 1.4.

2. A modification of Oliver’s construction

Definition. Let p be a prime, S a finite p-group and N ⊴ S a normal subgroup. A Y -series in S for N is a sequence 1 = Y_{0} ≤ Y_{1} ≤ ⋯ ≤ Y_{n} = N of normal subgroups Y_{i} ⊴ S such that

[Ω_{1}(C_{S}(Y_{i−1})), Y_{i}; 2] = 1

holds for each 1 ≤ i ≤ n. The unique largest normal subgroup N which admits a Y -series is denoted Y(S).

Lemma 2.1. Let p be a prime and S a finite p-group.

(1) If 1 = Y_{0} ≤ Y_{1} ≤ ⋯ ≤ Y_{n} = N is a Y -series in S and M ⊴ S also admits a Y -series, then there is a Y -series for MN starting with Y_{0}, …, Y_{n}.

(2) There is indeed a unique largest normal subgroup Y(S) admitting a Y - series. Moreover, Y(S) is characteristic in S.

(3) Every abelian normal subgroup of S lies in Y(S). In particular, Y(S) is centric in S.

(4) If p is odd then Y(S) ≤ 𝔛(S). If p = 3 then Y(S) = 𝔛(S).

(5) If p = 2 then Y(S) = S.

Proof. (1) See the corresponding proof for Q-series, at the bottom of p. 334

in [10].

(2) Unique existence follows from the first part. It is characteristic because it is unique.

(3) The proof of [10, Lemma 3.2] works for Y(S) as well, even for p = 2.

(4) If p ≥ 3 then every Y -series is a Q-series. If p = 3 then the two notions coincide.

(5) If not, then Y(S) < S. Choose T > Y(S) such that T/Y(S) is cyclic of order 2 and contained in Ω_{1}(Z(S/Y(S))). Then T ⊴ S, and we must have [Ω_{1}(Z(Y(S))), T; 2] ≠ 1, since Y(S) is centric and the Y -series for Y(S) cannot be extended to include T.

As Y(S) has index 2 in T, we have t^{2} ∈ Y(S) for each t ∈ T. Hence by Eqn (2.2) of [5] we have

[Ω_{1}(Z(Y(S))), t; 2] = [Ω_{1}(Z(Y(S))), t^{2}] = 1 ,

a contradiction. □

Example. Oliver remarks in [10, p. 335] that 𝔛(S) = C_{S}(Ω_{1}(S)) for any finite 2-group S. So if S is dihedral of order 8, then 𝔛(S) = Z(S) is cyclic of order 2, whereas Y(S) = S has order 8. Hence 𝔛(S) < J(S) = Y(S) = S in this case.

Example. Let S = C_{5}^{3} ⋊ C_{5} be the semidirect product in which the cyclic group on top acts via a (3 × 3) Jordan block (with eigenvalue 1) on the rank three elementary abelian on the bottom. Then C_{5}^{3} = J(S) = Y(S) ⪇ 𝔛(S) = S.

Remark. Let p be an odd prime. The proof of [4, Lemma 2.3] also shows that for every p-group G there is a p-group S with S/Y(S) ≅ G.

3. Orthogonality

Notation. Suppose that p is an odd prime, that G is a non-trivial p-group, and that V is a faithful (right) 𝔽_{p}G-module. Let I be the kernel of the structure map 𝔽_{p}G → End(V). Recall from [5, Section 2] that [v, g] = v(g − 1) for all v ∈ V, g ∈ G. Hence [V, g, h] = 0 if and only if (g − 1)(h − 1) ∈ I.

For elements g, h ∈ G we write g ⊥_{V} h or simply g ⊥ h if [g, h] = 1 and (g − 1)(h − 1) ∈ I. Note that ⊥_{V} is a symmetric relation on G. We write

g^{⊥} := {h ∈ G | g ⊥ h} .

Observe that 1 ≠ g ∈ G is quadratic if and only if g ⊥ g.

Lemma 3.1. Let V be a faithful 𝔽_{p}G-module, where G is a non-trivial p-group. For any g, h, x ∈ G we have:

(1) g is quadratic if and only if g ∈ g^{⊥}.

(2) The relation h ∈ g^{⊥} is symmetric.

(3) The set g^{⊥} is a subgroup of C_{G}(g).

(4) For any integer r coprime to p, we have that (g^{r})^{⊥} = g^{⊥} and therefore

g^{r} quadratic ⟺ g quadratic.

(5) Assume that p is odd. If g, h are both quadratic and [g, h] = 1 then

gh is quadratic ⟺ g ⊥ h .

(6) Assume that p is odd. If g is quadratic and [g, g^{x}] = 1 then

g ⊥ g^{x} ⟺ [g, x] is quadratic.

Proof. Parts (1) and (2) are clear. Part (3): Obviously g ⊥ 1. If g ⊥ h and g ⊥ k, then

(g − 1)(hk − 1) = (g − 1)(h − 1) + h(g − 1)(k − 1)

and so g ⊥ hk. Inverses follow, since G is finite.

Part (4): If g = 1 then there is nothing to prove. If g ≠ 1 then r is a unit modulo the order of g. So it suffices to show the inclusion g^{⊥} ≤ (g^{r})^{⊥}. As g^{r} only depends on the residue class of r modulo the order of g, we may assume that r ≥ 1. Then

g^{r} − 1 = a(g − 1) for a = ∑_{i=0}^{r−1} g^{i} .

So if (g − 1)(h − 1) lies in the kernel I of the representation, then so does (g^{r} − 1)(h − 1).

Part (5): From (gh − 1) = (g − 1) + g(h − 1), we get that

(gh − 1)^{2} = (g − 1)^{2} + 2g(g − 1)(h − 1) + g^{2}(h − 1)^{2}

because [g, h] = 1. So since 2g is invertible and (g − 1)^{2}, (h − 1)^{2} ∈ I, we see that (gh − 1)^{2} ∈ I if and only if (g − 1)(h − 1) ∈ I.

Part (6): By (4) we see that g^{−1} is quadratic if and only if g is; and by (3) we have g^{−1} ∈ (g^{x})^{⊥} if and only if g ∈ (g^{x})^{⊥}. But [g, x] = g^{−1}g^{x}. Now apply (5). □

Corollary 3.2. Let p be an odd prime, G a non-trivial p-group, and V a faithful 𝔽_{p}G-module. If E = ⟨g_{1}, g_{2}, …, g_{r}⟩ ≤ G is elementary abelian then the following three statements are equivalent:

(1) Every element 1 ≠ g ∈ E is quadratic.

(2) g ⊥ h for all g, h ∈ E.

(3) g_{i} ⊥ g_{j} for all i, j ∈ {1, …, r}.

Notation. Recall that an elementary abelian subgroup E ≤ G is called quadratic (for V) if it satisfies the equivalent conditions of Corollary 3.2.

Proof. Clearly the second statement implies the first. The first implies the third, by Lemma 3.1 (5). Now assume the third condition holds. Then E ≤ (g_{i})^{⊥} for every i, since (g_{i})^{⊥} is a group (Lemma 3.1 (3)) and contains each g_{j}. So g ⊥ g_{i} for each i and for each g ∈ E. Hence g_{i} ∈ g^{⊥} for every i. Therefore E ≤ g^{⊥}. So g ⊥ h for all g, h ∈ E, and the second condition holds. □

Lemma 3.3. Suppose that g, h ∈ G with g ≠ 1 and g ⊥ h. Suppose further that C is a subgroup of C_{G}(h) which contains g and has cyclic centre. Then

Ω_{1}(Z(C)) ≤ h^{⊥} .

Proof. We proceed by induction on the smallest integer r ≥ 1 with g ∈ Z_{r}(C). If r = 1 then Ω_{1}(Z(C)) ≤ ⟨g⟩, and we are done since ⟨g⟩ ≤ h^{⊥}. If g ∈ Z_{r+1}(C), then there is an x ∈ C such that 1 ≠ y := [g, x] ∈ Z_{r}(C). By induction it suffices to show that y ⊥ h. Since x ∈ C we have [h, x] = 1 and therefore x^{−1}(g−1)(h−1)x = (g^{x}−1)(h−1), showing that g^{x} ⊥ h. So y ∈ ⟨g, g^{x}⟩ ≤ h^{⊥}. □

We close this section by recalling without proof a key lemma from [4].

Lemma 3.4 (Lemma 4.1 of [4]). Suppose that p is an odd prime, that G is a non-trivial p-group, and that V is a faithful (right) 𝔽_{p}G-module. Suppose that A, B ∈ G are such that C := [B, A] is a nontrivial element of C_{G}(A, B). If B is quadratic, then so is C. □

4. Weakly closed subgroups

This section is related to work of Chermak and Delgado [2], especially the case α = 1 of their Theorem 2.4.

Notation. For the sake of brevity we will say that an abelian subgroup A of G is weakly closed if A is weakly closed in C_{G}(A) with respect to G. That is, A ≤ G is weakly closed if [A, A^{g}] ≠ 1 holds for every G-conjugate A^{g} ≠ A.

Remark. Every maximal elementary abelian subgroup M of G is weakly closed, since if M^{g} ≠ M but [M, M^{g}] = 1 then ⟨M, M^{g}⟩ is elementary abelian and strictly larger. Hence every elementary abelian subgroup is contained in a weakly closed one. If the normal closure of E is non-abelian, then every weakly closed elementary abelian subgroup containing E is non-normal.

Remark 4.1. Note that if A ≤ G is an abelian subgroup and g ∈ G then

[A, A^{g}] = 1 ⟺ [A, [A, g]] = 1 ⟺ [g, A, A] = 1 .

In particular, if A is weakly closed (in our sense), then Z_{2}(G) ≤ N_{G}(A): for if g ∈ Z_{2}(G) then [A, g] ≤ Z(G) and therefore [A, [A, g]] = 1. Hence [A, A^{g}] = 1 and so A = A^{g}, that is g ∈ N_{G}(A).

Remark 4.2. Using GAP [3], the authors have constructed the following examples:

• The Sylow 3-subgroup G of the symmetric group S_{27} contains a rank four weakly closed elementary abelian E with E ∩ Z(G) = 1.

• The Sylow 3-subgroup G of the symmetric group S_{81} contains a rank six

weakly closed elementary abelian E with E ∩ Z_{2}(G) = 1. The GAP code is available from the first author on request.

Lemma 4.3. Suppose that G is a finite p-group and that the elementary abelian subgroup E of G is weakly closed. Then N_{G}(E) = N_{G}(C_{G}(E)). So if E is not central in G then N_{G}(E) ⪈ C_{G}(E).

Proof. N_{G}(E) always normalizes C_{G}(E). If x ∈ G normalizes C_{G}(E) then as E ≤ C_{G}(E) we have that E^{x} ≤ C_{G}(E) and therefore [E, E^{x}] = 1. So E^{x} = E, for E is weakly closed. Hence x ∈ N_{G}(E). Last part: G is a nilpotent group. So if C_{G}(E) is a proper subgroup of G, then it is properly contained in its normalizer. □

Notation. Let G be a finite group, H ≤ G a subgroup, and V a faithful 𝔽_{p}Gmodule. Following Meierfrankenfeld and Stellmacher [9] we set

j_{H}(V) := \frac{|H| · |C_{V} (H)|}{|V |} ∈ ℚ .

This means that an elementary abelian subgroup E of G is an offender if and only if E ≠ 1 and j_{E}(V) ≥ 1 = j_{1}(V).

Lemma 4.4 (Lemma 2.6 of [9]). Let G be a finite group and V a faithful 𝔽_{p}Gmodule. Let H, K be subgroups of G with ⟨H, K⟩ = HK. Then

j_{HK}(V)j_{H∩K}(V) ≥ j_{H}(V)j_{K}(V) ,

with equality if and only if C_{V} (H ∩ K) = C_{V} (H) + C_{V} (K). □

Remark. As some readers may find the article [9] by Meierfrankenfeld and Stellmacher hard to obtain, we reproduced the proof of this result in our earlier paper [5, Lemma 3.1] – though unfortunately we accidentally omitted the necessary assumption that ⟨H, K⟩ = HK.

Recall that a faithful 𝔽_{p}G-module V is called an F -module if it has at least one offender.

Proposition 4.5. Suppose that the faithful 𝔽_{p}G-module V is an F -module. Set

j_{0} = max{j_{E}(V) | E an offender} .

Then there is a weakly closed quadratic offender E with j_{E}(V) = j_{0}.

Moreover if D ≤ G is any offender with j_{D}(V) = j_{0}, then there is such an E which is a subgroup of the normal closure of D.

Proof. Let D be an offender with j_{D}(V) = j_{0}. Then D has a subgroup C ≤ D which is minimal by inclusion amongst the offenders with j_{C}(V) = j_{0}. By maximality of j_{0}, the version of Timmesfeld’s replacement theorem in [5, Theorem 4.1] then tells us that C is a quadratic offender.

Suppose first that j_{0} > 1. We shall show that C is weakly closed. If not, then A := ⟨C, C^{g}⟩ is elementary abelian for some g ∈ G with C^{g} ≠ C. Then j_{A}(V) ≤ j_{0} by maximality of j_{0}. And since j_{0} > 1 = j_{1}(V), we have that j_{C∩C^{g}}(V) < j_{0} by maximality of j_{0} and minimality of C. But by Lemma 4.4, this means that

j_{0}^{2} > j_{A}(V)j_{C∩C^{g}}(V) ≥ j_{C}(V)j_{C^{g}}(V) = j_{0}^{2} ,

a contradiction. So E = C ≤ D has the required properties.

Now suppose that j_{0} = 1. Let g ∈ G be such that C^{g} ≠ C and [C, C^{g}] = 1. As ⟨C, C^{g}⟩ and C ∩ C^{g} both have j ≤ 1, Lemma 4.4 means that both have j = 1, and (by equality) C_{V} (C ∩C^{g}) = C_{V} (C)+C_{V} (C^{g}). Furthermore, minimality of C means that C ∩ C^{g} = 1, and so C_{V} (C) + C_{V} (C^{g}) = V. As in the proof of [5, Lemma 4.3], this means that [V, C, C^{g}] = 0. Let T ⊆ G be a subset maximal with respect to the condition that E := ⟨C^{g} | g ∈ T ⟩ is abelian (and therefore elementary abelian). Applying the above argument to any g, h ∈ T with C^{g} ≠ C^{h} we deduce that [V, C^{g}, C^{h}] = 0. So since each C^{g} is quadratic, we deduce that E is quadratic too. Finally, a repeated application of Lemma 4.4 coupled with the fact that j never exceeds 1 tells us that j_{E}(V) = 1 too. But by construction, E is weakly closed and contained in the normal closure of D. □

It follows from Proposition 4.5 that Conjecture 1.4 holds for (G, V) if the following conjecture does.

Conjecture 4.6. Let p be a prime, G a finite p-group, and V a faithful 𝔽_{p}Gmodule. If there is an elementary abelian subgroup 1 ≠ E ≤ G which is both quadratic on V and weakly closed in C_{G}(E) with respect to G, then there are quadratic elements in Ω_{1}(Z(G)).

We establish Theorem 1.6 by demonstrating that the Sylow subgroups of the symmetric groups satisfy Conjecture 4.6.

5. An inductive hypothesis

We now present the inductive hypothesis (Hypothesis 5.2) that will be used in Section 6 to verify Conjecture 1.4 for an iterated wreath product. First though, we need a few auxilliary lemmas.

Lemma 5.1. Suppose that G is a direct product of the form G = H × P, where H and P are p-groups. Let E ≤ G be an elementary abelian subgroup which is weakly closed (in C_{G}(E) with respect to G). Set

F = {g ∈ P | ∃h ∈ H (h, g) ∈ E} ≤ P ,

and set N = N_{P} (F). Then the following hold:

(1) F is weakly closed (in C_{P} (F) with respect to P).

(2) 1 × [F, N] ≤ E.

(3) If E ≰ H × Z(P), then [F, N] ≠ 1 and therefore E ∩ (1 × P) ≠ 1.

Proof. (1) If x ∈ P and F^{x} ≠ F then E^{(1,x)} ≠ E and therefore [E, E^{(1,x)}] ≠ 1. But [E, E^{(1,x)}] = 1 × [F, F^{x}]. So [F, F^{x}] ≠ 1.

(2) Let f ∈ F and n ∈ N. Pick h ∈ H such that (h, f) ∈ E. Since [F, F^{n}] = [F, F] = 1, we have [E, E^{(1,n)}] = 1 by the proof of the first part. Hence E^{(1,n)} = E, and therefore (1, [f, n]) = [(h, f), (1, n)] ∈ E.

(3) F is weakly closed by the first part, and non-central by assumption. Hence 1 ≠ [F, N] ≤ F by Lemma 4.3. Done by the second part. □

Hypothesis 5.2. Let P be a p-group with the following properties:

• P is nonabelian with cyclic centre.

• Suppose that H is a p-group and that V is a faithful 𝔽_{p}G-module, where G = H×P. If E is a weakly closed quadratic elementary abelian subgroup of G which satisfies E ≰ H × Z(P), then the subgroup 1 × Ω_{1}(Z(P)) of Z(G) is quadratic.

At the end of this section (Corollary 5.7) we give some first examples of groups that satisfy this hypothesis. In the next section (Corollary 6.6) we shall show that the Sylow p-subgroups of S_{p^{n}} also satisfy it. First though we explain the significance of the hypothesis for Oliver’s conjecture.

Lemma 5.3. Suppose that the finite p-group G is a direct product G = ∏_{r=1}^{n} H_{r}, where each H_{r} is either abelian or satisfies Hypothesis 5.2. Then G satisfies Conjectures 1.4 and 4.6.

Proof. For 1 ≤ r ≤ n, we shall denote by K_{r} the product ∏_{i≠r} H_{i}. Hence G = K_{r} × H_{r} for each r. Note that Z(G) = ∏_{r} Z(H_{r}).

Conjecture 4.6 implies Conjecture 1.4 by Proposition 4.5, and so we assume that there is a weakly closed quadratic elementary abelian subgroup E of G with E ≠ 1. If E ≤ Z(G) then every element 1 ≠ g ∈ E is a quadratic element of Ω_{1}(Z(G)). If E ≰ Z(G) = ∏_{r} Z(H_{r}) then for some 1 ≤ r ≤ n we have E ≰ K_{r} × Z(H_{r}). It follows that H_{r} cannot be abelian, and so by assumption it satisfies Hypothesis 5.2: which means that the subgroup 1 × Ω_{1}(Z(H_{r})) of Ω_{1}(Z(G)) is quadratic. □

We are particularly interested in wreath products of the form G = P ≀C_{p}, where P is a p-group. Recall that this means that G is the semidirect product G = P^{p}⋊C_{p}, where the C_{p} on top acts on the base P^{p} = ∏_{1}^{p} P by permuting the factors cyclically. In particular, this means that we may view P^{p} as a subgroup of G.

We start with a minor diversion. The following result generalizes [6, Satz III.15.4a)] slightly and is presumably well known. It is needed for the proof of Theorem 1.8.

Lemma 5.4. Suppose that p is an odd prime and that P is a p-group such that J(P) is elementary abelian. Then J(P ≀ C_{p}) is elementary abelian too. In particular, if P ≠ 1 then J(P ≀ C_{p}) is the copy of J(P)^{p} in the base subgroup P^{p} ≤ P ≀ C_{p}.

Proof. If P = 1 then P ≀ C_{p} = C_{p} is elementary abelian and we are done. So we may assume that P ≠ 1, which means that r ≥ 1, where r is the p-rank of P. Note that J(P) ≅ C_{p}^{r} by assumption.

Consider the base subgroup P^{p}, which has index p in P ≀ C_{p} = P^{p} ⋊ C_{p}. Since J(G_{1} × G_{2}) = J(G_{1}) × J(G_{2}), we see that J(P^{p}) = J(P)^{p}, which is elementary abelian of rank pr. So if J(P ≀ C_{p}) is not elementary abelian then P^{p} ⋊ C_{p} must contain an elementary abelian subgroup E of rank ≥ pr with E ≰ P^{p}. We set E′ = E ∩ P^{p}. Since P^{p} has index p in P ≀ C_{p}, it follows that |E: E′| = p and therefore that E′ ≤ P^{p} is elementary abelian of rank ≥ pr−1. Furthermore there is an element g ∈ (P ≀ C_{p}) \ P^{p} with E = ⟨E′, g⟩ and therefore [E′, g] = 1.

We split P^{p} as P^{p} = A × B, where A, B ≤ P^{p} are given by

A = {(a_{1}, 1, …, 1) | a_{1} ∈ P }  B = {(1, a_{2}, …, a_{p}) | a_{i} ∈ P } .

Note that E′ ∩ B must be trivial: for no non-trivial element of B can commute with g ∈ (P ≀ C_{p}) \ P^{p}. Hence the projection of E′ onto A is injective. As A ≅ P has rank r, this means that pr − 1 ≤ r. This is impossible with p odd and r ≥ 1. □

Now we derive a property of wreath products which will be useful for verifying the hypothesis.

Lemma 5.5. Suppose that G is the wreath product P ≀ C_{p}, where P is a finite p-group. Suppose that E is a weakly closed elementary abelian subgroup of G and that either of the following properties is satisfied:

(1) E is not contained in the base subgroup P^{p} of G.

(2) Z(P) is cyclic, E is non-central, and E lies in an abelian normal subgroup

of G. Then Ω_{1}(Z(G)) ≤ [E, N_{G}(E)].

Proof. The group Ω_{1}(Z(G)) is cyclic of order p. It is generated by (z, …, z) ∈ P^{p} for any element z ∈ Ω_{1}(Z(P)) with z ≠ 1.

Suppose first that (2) holds. As E lies in an abelian normal subgroup of G and is weakly closed, we deduce that E ⊴ G. So [E, N_{G}(E)] = [E, G] is normal too. Moreover, [E, G] is nontrivial, because E is non-central. Since |Ω_{1}(Z(G))| = p, it follows that Ω_{1}(Z(G)) ≤ [E, G] = [E, N_{G}(E)].

Now suppose that (1) holds. Pick an element z ∈ Ω_{1}(Z(P)) with z ≠ 1. It suffices to show that (z, …, z) ∈ [E, N_{G}(E)]. We start by showing that c = (1, z, z^{2}, …, z^{p−1}) satisfies c ∈ Z_{2}(G).

If g ∈ P^{p} then [g, c] = 1, since c ∈ Z(P^{p}). If g ∈ G \ P^{p} then g = hσ, where h ∈ P^{p} and σ is a cyclic permutation of the p factors of P^{p}. Hence

[g, c] = [σ, c] = (c^{σ})^{−1}c = (z, z, …, z)^{r} ∈ Z(G) for some 1 ≤ r ≤ p − 1.

Therefore c ∈ Z_{2}(G) as claimed. By Remark 4.1 if follows that c ∈ N_{G}(E).

Now by assumption E ≰ P^{p}, and so E contains some g = hσ in G \ P^{p}. So replacing g by a suitable power if necessary, we may suppose that r = 1, and so (z, …, z) = [g, c] ∈ [E, c] ≤ [E, N_{G}(E)]. □

The following lemma gives one way of verifying that the hypothesis is satisfied.

Lemma 5.6. Let P be a non-abelian p-group with cyclic centre. Suppose that for every non-central weakly closed elementary abelian subgroup F of P we have that [F, N_{P} (F)] ∩ Z_{2}(P) ≠ 1. Then P satisfies Hypothesis 5.2.

Proof. Let G = H × P for a p-group H, and let E be a weakly closed quadratic elementary abelian subgroup of G with E ≰ H × Z(P). We need to show that 1 × Ω_{1}(Z(P)) is quadratic.

Consider the elementary abelian subgroup F ≤ P defined by F = {g ∈ P | ∃h ∈ H (h, g) ∈ E}. Then F is weakly closed (in C_{P} (F) with respect to P) by Lemma 5.1 (1). Moreover, F ≰ Z(P), since [F, N_{P} (F)] ≠ 1 by Lemma 5.1 (3). So by assumption we may pick g ∈ Z_{2}(P) ∩ [F, N_{P} (F)] with g ≠ 1. Since g ∈ [F, N_{P} (F)], we have (1, g) ∈ E by Lemma 5.1 (2). Hence (1, g) is quadratic and of order p. If g ∈ Z(P) then we are done, since Z(P) is cyclic.

If g ∈ Z_{2}(P) \ Z(P) then 1 ≠ [g, x] generates Ω_{1}(Z(P)) for some x ∈ P. But then (1, [g, x]) = [(1, g), (1, x)] is quadratic by Lemma 3.4. □

Corollary 5.7. Let p be a prime and P a finite p-group. Assume that either of the following holds:

(1) P is a wreath product of the form P = C_{p^{r}} ≀ C_{p} for r ≥ 1;

(2) Z(P) is cyclic, and P has nilpotence class two or three. Then Hypothesis 5.2 holds for P.

Proof. In both cases, P is nonabelian and has cyclic centre. Let F ≤ P be a non-central elementary abelian subgroup which is weakly closed in C_{P} (F) with respect to P. By Lemma 5.6 it suffices to show that Z_{2}(P) ∩ [F, N_{P} (F)] ≠ 1.

(1) We prove that Ω_{1}(Z(P)) ≤ [F, N_{P} (F)]. The case F ≰ C_{p^{r}}^{p} is covered by Lemma 5.5 (1). Since C_{p^{r}}^{p} is normal abelian, the case F ≤ C_{p^{r}}^{p} is accounted for by Lemma 5.5 (2).

(2) Since F is weakly closed and non-central we have C_{P} (F) ⪇ N_{P} (F) by Lemma 4.3. Hence [F, N_{P} (F)] ≠ 1. And as the (nilpotence) class of P is at most three we have [F, N_{P} (F), P] ≤ Z(P), whence [F, N_{P} (F)] ≤ Z_{2}(P). □

6. Wreath products

We saw in Lemma 5.3 that Conjecture 1.4 holds for the direct product H_{1} ×⋯× H_{n} if each p-group H_{r} is abelian or satisfies Hypothesis 5.2. In Corollary 5.7 we saw our first examples of groups which satisfy the hypothesis. We now demonstrate that if P satisfies the hypothesis, then so does the wreath product P ≀ C_{p}.

This is the key step in proving that the Sylow subgroups of a symmetric group satisfy Conjecture 1.4.

Proposition 6.1. If the p-group P satisfies Hypothesis 5.2 then so does the wreath product Q = P ≀ C_{p}.

Proof. Certainly Q is nonabelian with cyclic centre. Let H be a p-group, and V a faithful module for G = H × Q. Recall that Q is a semidirect product, with base subgroup K := ∏_{1}^{p} P on which C_{p} acts by permuting the factors cyclically. Let E ≤ G be a quadratic elementary abelian subgroup which is weakly closed in C_{G}(E) with respect to G, and also satisfies E ≰ H × Z(Q). We have to show that 1 × Ω_{1}(Z(Q)) is quadratic.

Let F = {g ∈ Q | ∃h ∈ H (h, g) ∈ E}. Then F is a non-central elementary abelian subgroup of Q; and by Lemma 5.1 it is weakly closed (in C_{Q}(F) with respect to Q).

Suppose first that E ≰ H × K. Then F ≰ K, and so by Lemma 5.5 (1) we have Ω_{1}(Z(Q)) ≤ [F, N_{Q}(F)]. The same thing happens if E ≤ H × Z(K): for then F ≤ Z(K), which is an abelian normal subgroup of Q. Hence Ω_{1}(Z(Q)) ≤ [F, N_{Q}(F)] by Lemma 5.5 (2). In both cases we then deduce from Lemma 5.1 (2) that 1 × Ω_{1}(Z(Q)) lies in E and is therefore quadratic.

So from now on we may assume that E ≤ H × K but E ≰ H × Z(K). That is, F is a subgroup of K = ∏_{1}^{p} P, but F ≰ Z(K) = Z(P)^{p}. Define a subset T (F) ⊆ {1, 2, …, p} by

i ∈ T (F) ⟺ F ≰ P × P × ⋯ × Z(P) × ⋯ × P ,

where the factor Z(P) occurs in the ith copy of P. Since F ≰ Z(P)^{p}, we have T (F) ≠ ∅.

For 1 ≤ i ≤ p we define subgroups L_{i}, P (i) ≤ H × K as follows:

L_{i} = H × (P × P × ⋯ × 1 × ⋯ × P) P (i) = 1 × (1 × 1 × ⋯ × P × ⋯ × 1) ,

where the brackets enclose K and the one distinguished factor occurs in the ith factor of K = P^{p}. Then H × K = L_{i} × P (i) for each i. Observe that

i ∈ T (F) ⟺ E ≰ L_{i} × Z(P (i)) .  (6.2)

Pick 1 ≠ ζ ∈ Ω_{1}(Z(Q)). We need to show that (1, ζ) ∈ Z(G) is quadratic. Note that ζ = (z, z, …, z) ∈ K for some 1 ≠ z ∈ Ω_{1}(Z(P)). That is, ζ = z_{1}z_{2} ⋯ z_{p}, where for 1 ≤ i ≤ p we define z_{i} ∈ Z(K) by

z_{i} = (1, …, z, …, 1) (z at the ith place).

Now, the group P (i) ≅ P satisfies Hypothesis 5.2 by assumption. Applying the hypothesis to the direct product group L_{i} ×P (i) = H ×K, we see from Eqn (6.2) that (1, z_{i}) is quadratic for every i ∈ T (F). As T (F) is nonempty and all the z_{i} are conjugate in Q under the cyclic permutation of the factors of K = P^{p}, it follows that every (1, z_{i}) is quadratic.

From Lemma 5.1 (3) and Eqn (6.2) it follows that for every i ∈ T (F) there is some 1 ≠ g_{i} ∈ P such that (1, k_{i}) ∈ E for k_{i} = (1, …, g_{i}, …, 1) ∈ K. Now suppose that i, j ∈ T (F) are distinct. Then (1, k_{i}) ⊥ (1, k_{j}), as they both lie in E. Applying Lemma 3.3 with C = P (i), we have that (1, z_{i}) ⊥ (1, k_{j}). Applying this lemma a second time, but now with C = P (j), we deduce that

(1, z_{i}) ⊥ (1, z_{j})  (6.3)

holds for all i, j ∈ T (F). We have already seen that it always holds for i = j.

As Q is the wreath product P ≀ C_{p}, we may pick an element x ∈ Q \ K such that (h_{1}, …, h_{p})^{x} = (h_{p}, h_{1}, …, h_{p−1}) holds for all (h_{1}, …, h_{p}) ∈ P^{p} = K. Then T (F^{x}) = σ(T (F)), where σ is the p-cycle (1 2 … p). Hence T (F^{x^{m}}) = σ^{m}(T (F)). We claim that

∀ 0 ≤ m ≤ p − 1 σ^{m}(T (F)) ∩ T (F) ≠ ∅ .  (6.4)

For if σ^{m}(T (F)) ∩ T (F) = ∅, then [F^{x^{m}}, F] = 1 by definition of T (F). But F is weakly closed, and so [F^{x^{m}}, F] = 1 implies that F^{x^{m}} = F. But then σ^{m}(T (F)) = T (F^{x^{m}}) = T (F), and so σ^{m}(T (F)) ∩ T (F) = T (F) ≠ ∅, a contradiction.

So Eqn (6.4) is proved. We may rephrase it as follows:

∀ 0 ≤ m ≤ p − 1 ∃ a, b ∈ T (F) b ≡ a + m (mod p) .  (6.5)

Now pick any i, j ∈ {1, 2, …, p}. By Eqn (6.5) there are a, b ∈ T (F) with b − a ≡ j − i (mod p). Define r to be the integer 0 ≤ r < p such that r ≡ i − a (mod p). Then

i = a + r (mod p)  and  j = b + r (mod p) .

Hence

(1, z_{i}) = (1, z_{a})^{(1,x^{r})}  and  (1, z_{j}) = (1, z_{b})^{(1,x^{r})} .

But (1, z_{a}) ⊥ (1, z_{b}), as this is one of the cases for which we have already demonstrated Eqn (6.3). So we deduce that (1, z_{i}) ⊥ (1, z_{j}): that is, Eqn (6.3) holds for all i, j. From ζ = z_{1}z_{2} ⋯ z_{p} we therefore deduce by Corollary 3.2 that (1, ζ) ⊥ (1, ζ), that is (1, ζ) is quadratic, as required. □

Corollary 6.6. The Sylow p-subgroups of the symmetric group S_{p^{n}} satisfy Hypothesis 5.2 for every n ≥ 2.

Proof. By induction on n. It is well known that the Sylow p-subgroups P_{n} of S_{p^{n}} are isomorphic to the iterated wreath product C_{p} ≀ C_{p} ≀ ⋯ ≀ C_{p}, with n copies of C_{p}. So P_{2} = C_{p} ≀ C_{p} satisfies the hypothesis by Corollary 5.7 (1). If P_{r} satisfies the hypothesis, then so does P_{r+1} = P_{r} ≀ C_{p} by Proposition 6.1. □

7. Deepest commutators

In this and the subsequent section we assemble the tools needed for the proof of Theorems 1.7 and 1.9.

Notation. Let G be a p-group and g ∈ G a non-central element. Let r_{0} = max{r | ∃k ∈ K_{r}(G): [g, k] ≠ 1}. This exists by nilpotence, since g is not central.

If k ∈ K_{r_{0}}(G) and [g, k] ≠ 1, then we call [g, k] a deepest commutator of g.

Remark. Observe that r_{0}(g) = r_{0}(g^{x}) for every x ∈ G.

Lemma 7.1. Let G be a p-group and g ∈ G a non-central element.

(1) g has at least one deepest commutator, and [g, [g, k]] = 1 holds for every deepest commutator [g, k] of g.

(2) Suppose that k is any element of G such that y = [g, k] satisfies [g, y] = 1.

Then the order of y divides the order of g. In particular if y ≠ 1 and g has order p, then so does y.

Proof. Deepest commutators exist, and commute with g by maximality of r_{0}. If 1 ≠ y = [g, k] commutes with g, then g^{k} = gy and so (g^{p^{n}})^{k} = (gy)^{p^{n}} = g^{p^{n}}y^{p^{n}}. So if g^{p^{n}} = 1 then y^{p^{n}} = 1. □

The concept “deepest commutator of g” depends not only on the element g, but also on the ambient group G. When we need to stress the group G we shall use the phrase “deepest commutator in G”.

Notation. Let G be a p-group, g an element of G, and N a normal subgroup of G. If y, x ∈ L := C_{G}(g)N are such that y = [g, x] is a deepest commutator of g in L, then we call y a (G, N)-locally deepest commutator of g. A locally deepest commutator is a (G, N)-locally deepest commutator for some N.

Remark. Since N is normal, L is a group. By the existence of deepest commutators, g has a (G, N)-locally deepest commutator if and only if N ≰ C_{G}(g).

Lemma 7.2. If y is an (G, N)-locally deepest commutator of g ∈ G, then y = [g, x] for some x ∈ N.

Proof. We have that y = [g, x′] for some x′ ∈ C_{G}(g)N, and so x′ = ax for some a ∈ C_{G}(g), x ∈ N. But then g^{x′} = g^{x} and so [g, x] = y. □

8. Late and last quadratics

Throughout this section, G is a finite p-group and V a faithful 𝔽_{p}G-module. Recall that an element g ∈ G is called quadratic if its action on V has minimal polynomial (X − 1)^{2}.

Notation. Suppose that g ∈ G is quadratic. We call g late quadratic if every locally deepest commutator of g is non-quadratic. We call g last quadratic if every iterated commutator z = [g, h_{1}, h_{2}, …, h_{r}] with r ≥ 1 and h_{1}, …, h_{r} ∈ G is non-quadratic.

Remark. An element g ∈ G can only fail to have any locally deepest commutators if g ∈ Z(G), for it is only if g ∈ Z(G) that N ≤ C_{G}(g) holds for every N ⊴ G. In particular, every quadratic element in Z(G) is late quadratic.

Note that every last quadratic element is late quadratic. We shall see in Lemma 8.2 that each late quadratic element has elementary abelian normal closure in G. Hence all the non-trivial iterated commutators of a last quadratic element have order p.

Lemma 8.1. If G has quadratic elements, then it has both late quadratic and last quadratic elements.

Proof. As there are quadratic elements we may set

t_{0} := max{t | K_{t}(G) contains a quadratic element} .

Pick a quadratic element g ∈ K_{t_{0}}(G) and observe that every commutator lies in K_{t_{0}+1}(G). By the maximality of t_{0}, we see that g is both last and late quadratic.

□

Lemma 8.2. Every late quadratic element g of G lies in Ω_{1}(Z(Y(G))) and therefore satisfies Y(G) ≤ C_{G}(g).

Proof. We have already observed that all quadratic elements have order p. As Y(G) is a centric subgroup of G (Lemma 2.1), it suffices to show that each quadratic element g of G lies in C_{G}(Y(G)). Let 1 = Y_{0} ≤ ⋯ ≤ Y_{n} = Y(G) be a Y -series, so

Y_{r} ⊴ G and [Ω_{1}(C_{G}(Y_{r−1})), Y_{r}; 2] = 1 .  (8.3)

We show that g ∈ C_{G}(Y_{r}) by induction on r. If r = 0, this is clear. Suppose r ≥ 1 and g ∈ C_{G}(Y_{r−1}). If g ∉ C_{G}(Y_{r}) then by Lemma 7.2, g has a (G, Y_{r})-locally deepest commutator y = [g, x] with x ∈ Y_{r}. Then [y, g] = 1 by Lemma 7.1, and [y, x] = [g, x, x] = 1 by Eqn (8.3). So, as g is quadratic, y is too by Lemma 3.4. But y cannot be quadratic, since y is a locally deepest commutator of the late quadratic element g ∈ G. □

Remark 8.4. If we were attempting to prove Conjecture 1.4 by induction on |G| then we could assume that it holds for G/Y(G). By Theorem 1.5 this would mean that J(G) ≤ Y(G). It would then follow from Lemma 8.2 that every late quadratic element lies in Ω_{1}(Z(J(G))), the intersection of all the greatest rank elementary abelian subgroups of G.

Lemma 8.5. Let H ≤ G be the subgroup generated by all last quadratic elements. Then every quadratic element of G lies in C_{G}(H).

Proof. If not then [g, h] ≠ 1 for some elements g, h ∈ G, with g last quadratic and h quadratic. By nilpotency [g, h; r] ≠ 1 and [g, h; r + 1] = 1 for some r ≥ 1. Let k = [g, h; r − 1]. As the normal closure of g in G is abelian (Lemma 8.2), all the iterated commutators of g commute, and so [[k, h], k] = 1. Also [[k, h], h] = [g, h; r + 1] = 1. Lemma 3.4 therefore says that 1 ≠ [k, h] is quadratic, since h is. But [k, h] = [g, h; r] cannot be quadratic, for g is last quadratic. Contradiction.

□

9. Powerful p-groups

We use L. Wilson’s paper [14] as a reference on powerful p-groups. Recall from [14, Definition 1.3] that for an odd prime p, a finite p-group G is called powerful if G′ ≤ G^{p} := ⟨g^{p} | g ∈ G⟩. As in the proof of Proposition 4.5, recall that Timmesfeld’s Replacement Theorem tells us that if V is a faithful F -module for G, then there is an elementary abelian subgroup E of G which is a quadratic offender. Our first result improves [5, Theorem 6.2]. Proposition 9.1. Let p be an odd prime and G a finite p-group. If G′ ∩ Ω_{1}(G) is abelian then G satisfies Conjecture 1.4. Proof. Suppose that V is a faithful 𝔽_{p}G-module with no quadratic elements in Ω_{1}(Z(G)). We must show that there are no quadratic offenders. So suppose that the elementary abelian subgroup E ≤ G is a quadratic offender. Then [G′, E] ≠ 1 by [5, Theorem 1.5(2)]. Moreover, G′E ⊴ G. Hence G′E ∩ Ω_{1}(G) ⊴ G. By the Dedekind Lemma [7, X.3], we have G′E ∩ Ω_{1}(G) = E(G′ ∩ Ω_{1}(G)), since E ≤ Ω_{1}(G). Therefore Z(E(G′ ∩ Ω_{1}(G))) is an abelian normal subgroup of G, and so by [5, Theorem 1.5(1)] it cannot contain E. This means that G′ ∩ Ω_{1}(G) is not centralized by E. So there must be an a ∈ E such that [G′ ∩ Ω_{1}(G), a] ≠ 1. But then a ∈ Ω_{1}(G), and so N := G′⟨a⟩ ∩ Ω_{1}(G) = ⟨a⟩(G′ ∩ Ω_{1}(G)) is a normal subgroup of G. Since G′ ∩ Ω_{1}(G) is abelian, a well-known result (quoted as [5, Lemma 6.1]) says that the commutator subgroup of N = ⟨a⟩(G′ ∩Ω_{1}(G)) consists of commutators in a. But N ⊴ G, and by choice of a we have N′ ≠ 1. Hence N′ has nontrivial intersection with Ω_{1}(Z(G)). So 1 ≠ [a, x] ∈ Ω_{1}(Z(G)) for some x ∈ G′ ∩ Ω_{1}(G). Since a is quadratic, Lemma 3.4 says that [a, x] ∈ Ω_{1}(Z(G)) is too. This contradicts our assumption that there are no quadratic elements in Ω_{1}(Z(G)). So G satisfies the conjecture. □ Theorem 9.2. Let p be an odd prime. Every powerful p-group G satisfies Conjecture 1.4. Proof. We refer to Wilson’s paper [14]. His Theorem 4.7 says that G^{p} is powerful, and hence his Corollary 4.11 applied with P = G shows that Ω_{1}(G^{p}) is powerful. On the other hand, his Theorem 3.1 says that Ω_{1}(G) has exponent p, and so Ω_{1}(G^{p}) = Ω_{1}(G) ∩ G^{p}. So Ω_{1}(G) ∩ G^{p} is powerful and of exponent p, which means it must be abelian. As G′ ≤ G^{p}, it follows that Ω_{1}(G) ∩ G′ is abelian. So G satisfies the conjecture, by Proposition 9.1 above. □

Acknowledgements. We are grateful to Burkhard Külshammer for suggesting that our methods might also apply to GL_{n}(𝔽_{q}) in the coprime characteristic case.

We acknowledge travel funding from the Hungarian Scientific Research Fund OTKA, grant 77-476.

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Department of Mathematics, Friedrich-Schiller-Universität Jena, 07737 Jena, Germany

E-mail address: david.green@uni-jena.de

Department of Algebra, Budapest University of Technology and Economics, Budapest, Hungary

Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom