ALMOST ALL GENERALIZED EXTRASPECIAL p-GROUPS ARE RESISTANT

RADU STANCU^{∗}

Abstract. A p-group P is called resistant if, for any finite group G having P as a Sylow p-subgroup, the normalizer N_{G}(P) controls p-fusion in G. The aim of this paper is to prove that any generalized extraspecial p-group P is resistant, excepting the case when P = E × A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or extraspecial of order p^{3} and exponent p (when p is odd). This generalizes a result of Green and Minh.

1. Introduction

Let G be a finite group and H a subgroup of G. Two elements of H are said to be fused in G if they are conjugate in G but not in H. We are interested in p-groups P such that for any finite group G, having P as a Sylow p-subgroup, the p-fusion is controled only by the normalizer N_{G}(P) of P (that is any two elements of P which are fused in G are fused in N_{G}(P)). In fact that this is equivalent to the requirement that any such group G does not contain essential p-subgroups (Definition 2.2). Following the terminology suggested by Jesper Grodal, we will call such a group resistant.

In fact, by a theorem of Mislin [Mi], the notion of resistant group is equivalent to what Martino and Priddy [MP] call Swan group. We recall that P is a Swan group if, for any G as before, the mod − p cohomology ring H^{∗}(G) is isomorphic to the mod − p cohomology ring H^{∗}(N_{G}(P)).

In a recent preprint [GM], Green and Minh proved that almost all extraspecial p-groups are Swan groups. In our paper we find the same result for generalized extraspecial p-groups (Definition 3.1) and give a proof avoiding cohomological methods.

2. Essential Groups

Let F_{p}(G) be the Frobenius category of a finite group G. We recall that the objects in this category are the non-trivial p-subgroups of G and the morphisms are the group homomorphisms given by the conjugation by elements of G. For a subgroup H of G we denote by F_{p}(G)_{≤H} the full subcategory of F_{p}(G) containing the non-trivial p-subgroups of H.

A natural question is: What is the minimal information needed to completely characterise these morphisms? For a Sylow p-subgroup P of G, Alperin showed in [Al] that these morphisms are locally controled, i.e. by normalizers N_{G}(Q) for Q a subgroup of P. Nine years later Puig [Pu1] refined this and required Q to be an essential p-subgroup of G. In what follows we will give the definition and some basic properties of essential p-subgroups of G.

Date: September 12, 2001.

∗ This work is a part of a doctoral thesis in preparation at the University of Lausanne, under the supervision of Prof. Jacques Thévenaz.

Definition 2.1. We say that Q is p-centric if Q is a Sylow p-subgroup of QC_{G}(Q) or, equivalently, Z(Q) is a Sylow p-subgroup of C_{G}(Q).

In the literature [Th, p. 324], a p-centric subgroup is also called p-self-centra-lizing. Note that if Q is p-centric then C_{P} (Q) = Z(Q) for any Sylow p-subgroup P of G containing Q.

Consider now the Quillen complex S_{p}(H) of a finite group H whose vertices are the objects in F_{p}(H) and whose simplices are given by chains of groups ordered by inclusion.

Definition 2.2. We say that Q is an essential subgroup of G if the Quillen complex S_{p}(N_{G}(Q)/Q) is disconnected and C_{G}(Q) does not act transitively on the connected components.

One can find in [Th, Thm. 48.8] that

Proposition 2.3. Q is an essential p-subgroup of G if and only if Q is p-centric and S_{p}(N_{G}(Q)/QC_{G}(Q)) is disconnected.

The proof has been done in a more general case. In the terminology and notation of [Th, Thm. 48.8] it suffices to replace local pointed groups by p-subgroups, N_{>Q} by S_{p}(N_{G}(Q))_{>Q} and OG by G. In most of the proofs of this paper we will use this proposition as an alternative definition of essential subgroups. For g ∈ G we denote by ^{g}Q the conjugte by g of Q.

Definition 2.4. We say that a subgroup H of a group G controls p-fusion in G if (|G : H|, p) = 1 and for any g ∈ G and any Q, such that Q and ^{g}Q are contained in H, there exists h ∈ H and c ∈ C_{G}(Q) such that g = hc, or, equivalently, if the inclusion H ↪ G induces an equivalence of categories F_{p}(H) ≃ F_{p}(G).

The notions of control of fusion and essential p-subgroups are strongly linked. The next proposition shows one of the aspects of this link.

Proposition 2.5 (Pu1, Ch. IV, Prop 2). The normalizer N_{G}(P) controls p-fusion in G if and only if there are no essential p-subgroups in G.

The proof is based on the variant of Alperin’s theorem using essential p-subgroups (see for instance [Th, Thm 48.3]) and on the fact that the essential p-subgroups are preserved by any equivalence of categories.

3. Generalized Extraspecial Groups

From now on C_{n} will denote the cyclic group of order n.

Definition 3.1. A p-group P is called generalized extraspecial if its Frattini subgroup, Φ(P), has order p, Φ(P) = [P, P] ≃ C_{p} and Z(P) ≥ Φ(P). If, moreover, Z(P) = Φ(P), P is called extraspecial.

Lemma 3.2. Let P be a generalized extraspecial p-group. Then either Z(P) is isomorphic to Φ(P) × A and P is isomorphic to E × A, or Z(P) is isomorphic to C_{p^{2}} × A and E is isomorphic to (E ∗ C_{p^{2}}) × A, where E is an extraspecial p-group, A is an elementary abelian group and ∗ means central product.

Proof : As Φ(P) is a cyclic subgroup of order p, the centre Z(P) doesn’t admit more than one factor isomorphic to C_{p^{2}} in its decomposition in cyclic subgroups, and if this factor exists, it contains Φ(P). Let A to be an elementary abelian subgroup of Z(P) such that Z(P) ≃ Φ(P) × A, when there is no C_{p^{2}} factor in Z(P), and Z(P) ≃ C_{p^{2}} × A, otherwise. We have, in both cases, [P, P] ∩ A = 1 and [P, A] = 1 so A is a direct factor of P. It is then straight forward that the complement of A in P is isomorphic either to E or to E ∗ C_{p^{2}}.

Recall that for |P | = p^{3} we have that P is isomorphic either to (C_{p} × C_{p}) ×C_{p} (in this case we say that P is of order p^{3} and exponent p) or C_{p^{2}} ×C_{p}, for p odd, and either to the dihedral group D_{8} or the quaternion group Q_{8}, for p = 2.

Let β : P/Z(P) × P/Z(P) → Φ(P) defined by β(x̄, ȳ) = [x, y]. It is a bilinear non-degenerate symplectic form on U := P/Z(P) viewed as a vector space over F_{p}. We recall that an isotropic vector subspace of U with respect to β is a subspace on which β is identically zero. A maximal isotropic subspace of U has dimension equal to half of the dimension of U.

Lemma 3.3. Let Q be a p-centric subgroup of P. Then Q contains Z(P) and Q/Z(P) contains a maximal isotropic subspace of P/Z(P).

Proof A p-centric subgroup of P clearly contains the centre Z := Z(P) of P. Suppose that V := Q/Z(P), considered as vector space, does not contain a maximal isotropic subspace of U := P/Z(P) whith respect to β. This means that there exists u ∈ U \ V with β(u, x) = 0, ∀x ∈ V. By taking a representaive e of u in P we have e ∈ P \ Q and e commutes with all the elements of Q. So e ∈ C_{P} (Q) \ Z(Q) which is a contradiction with the fact that Q is p-centric.

4. Resistant Groups

Definition 4.1. A p-group P is called resistant if for any finite group G such that P is a Sylow p-subgroup of G, the normalizer N_{G}(P) controls p-fusion in G.

Here is now the main result of this paper.

Theorem 4.2. Let P be a generalized extraspecial p-group. Then P is resistant excepting the case when P = E ×A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or extraspecial of order p^{3} and exponent p (when p is odd).

Corollary 4.3. If P satisfies the conditions of the theorem then P is a Swan group.

Proof of Theorem 4.2: We will prove that the only cases where G contains essential p-subgroups are the exceptions of our theorem. Let Q be a proper p-centric subgroup of P. This forces Q to contain Z(P) and hence also Φ := Φ(P). Denote by R the subgroup of N := (N_{G}(Q) ∩ N_{G}(Φ))/C_{G}(Q) acting trivially on Φ and Q/Φ. We have that R centralizes the quotients of the central series 1 ◁ Φ ◁ Q so it is a normal p-subgroup [Gor, Thm. 5.3.2] of N. Now R contains P/Z(Q) as P acts trivially on Φ and Q/Φ. As P is a Sylow p-subgroup of G, this forces R = P/Z(Q), and thus R is the unique Sylow p-subgroup of N and thus S_{p}(N) is connected.

Assume that Q is essential. Then S_{p}(N_{G}(Q)/QC_{G}(Q)) is disconnected and therefore N_{G}(Q) ≠ N_{G}(Q) ∩ N_{G}(Φ). As the Φ(Q) is characteristic in Q and is contained in Φ we have that Φ(Q) is a proper subgroup of Φ hence trivial; this gives that Q is elementary abelian. Take x ∈ N_{G}(Q)\N_{G}(Φ). Now R = P/Q is not contained in (N_{G}(Q)∩N_{G}(^{x}Φ))/C_{G}(Q), otherwise N/C_{G}(Q) and (N_{G}(Q)∩N_{G}(^{x}Φ))/C_{G}(Q) would have the same Sylow p-subgroup R implying that P/Q = ^{x}(P/Q) and thus that x normalizes P. It follows that Φ = ^{x}Φ which is in contradiction with the choice of x. As^{x}Φ is a subgroup of P of order p, the vector subspace^{x}Φ/(Z(P)∩^{x}Φ) of P/Z(P) admits an orthogonal complement with respect to β which is either all P/Z(P) or a hyperplane. This gives that |P : C_{P} (^{x}Φ)| = 1 or p. If Q is a proper subgroup of C_{P} (^{x}Φ) then C_{P} (^{x}Φ) is non-abelian and therefore Φ = Φ(C_{P} (^{x}Φ)). Moreover ^{x^{−1}}(C_{P} (^{x}Φ)/Q) ⊂ (C_{N_{G}(Q)}(Φ)/Q) so, by Sylow’s theorem there exists c ∈ (C_{N_{G}(Q)}(Φ)/Q) such that ^{cx^{−1}}(C_{P} (^{x}Φ)/Q) ⊂ (C_{P} (Φ)/Q). This implies that cx^{−1}Φ = Φ which is equivalent to Φ = ^{x}Φ and we obtain once again a contradiction. Hence Q = C_{P} (^{x}Φ) and |P : Q| = p. We also have that Q/Z(P) is a maximal isotropic subspace of P/Z(P); it follows that |P : Z(P)| = p^{2}. Moreover C_{P} (^{x}Φ) is a proper subgroup of P so ^{x}Φ is not contained in Z(P) impling that Z(P) ≠ ^{x}Z(P). By the same type of arguments we can also prove that Φ is not contained in ^{x}Z(P).

Finally take A := Z(P) ∩ ^{x}Z(P). As |Q : Z(P)| = |Q : ^{x}Z(P)| = p and Z(P) ≠ ^{x}Z(P) we obtain that |Z(P) : A| = p so Q/A is isomorphic to C_{p} × C_{p}. Moreover A doesn’t contain Φ so, by Lemma 3.2, Z(P) ≃ Φ × A and P ≃ E × A where E is an extraspecial group of order p^{3}. First, as Q/A is isomorphic to C_{p}×C_{p}, E cannot be isomorphic to the quaternion group. Secondly we will prove that the case where E is isomorphic to C_{p^{2}} ×C_{p} also yields to a contradiction. The result is due to Glauberman [MP] but the proof we give, which is more elegant, is due to Jacques Thévenaz.

Let K := < P/Q, ^{x}(P/Q) >, which is isomorpic to a subgroup of Aut(Q/A) viewed as a subgroup of GL(2, F_{p}). As P/Q ≠ ^{x}(P/Q) they generate all SL(2, F_{p}), so SL(2, F_{p}) is a subgroup of K containing P/Q. Now P/Q is a Sylow p-subgroup of K and we will prove that the exact sequence 1 → Q/A → E → P/Q → 1 can be extended to an exact sequence 1 → Q/A → L → K → 1 and hence to an exact sequence 1 → Q/A → L′ → SL(2, F_{p}) → 1. To have this it suffices to verify [Br, pp.84-85] that the class h(E) determined by E in H^{2}(P/Q, Q/A) is K-stable, that is for any k ∈ K we have

res_{P/Q ∩^{k}(P/Q)}^{P/Q}h(E) = res_{P/Q ∩^{k}(P/Q)}^{^{k}(P/Q)}conj_{k}(h(E)) (∗) .

Here res is the restriction in cohomology and conj_{k} is the morphism induced by the conjugation by k in cohomology. If P/Q ≠ ^{k}(P/Q) then P/Q ∩ ^{k}(P/Q) = 1 and the relation (∗) is trivially is satisfied. Suppose that P/Q = ^{k}(P/Q). Take k̃ to be a representative of k in N_{G}(Q) that normalizes P. We have that k̃ induces the conjugation by k on Q and P/Q. So the conjugation by k̃ induces conj_{k} on H^{2}(P/Q, Q/A). Thus h(E) = conj_{k}(h(E)) and (∗) is again satisfied. Now, for E ≃ C_{p^{2}} ×C_{p}, h(E) is not trivial.

The contradiction comes from the fact that H^{2}(SL(2, F_{p}), Q/A) = 0, so the cohomology class h(E) induced by E in H^{2}(P/Q, Q/A) would be trivial. Indeed let

U := \{( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} )\} be a Sylow p-subgroup of SL(2, F_{p}). Write S := SL(2, F_{p}) and

N(U) := N_{SL(2,F_{p})}(U). The restriction to U in cohomology induces a monomorphism res_{U}^{S} : H^{2}(S, Q) → H^{2}(U, Q)^{N(U)} where H^{2}(U, Q)^{N(U)} are the fixed points under the natural action of N(U). Now U = < u > is a cyclic group so [Be, p. 60] its cohomology is

H^{2}(U, Q) = Q^{U}/{(∑_{i=0}^{p−1} u^{i})v|v ∈ Q} .

By a simple computation we obtain Q^{U} = < z >, where z is a generator of Φ(P) and {(∑_{i=0}^{p−1} u^{i})v|v ∈ Q} = 0 so H^{2}(U, Q) = < z >. As z is not fixed by N(U) we have H^{2}(U, Q)^{N(U)} = 0 and therefore H^{2}(S, Q) = 0.

We prove now that the remaning case, P = E × A with E either dihedral of order 8 (when p = 2) or extraspecial of order p^{3} and exponent p (when p is odd), is indeed an exception to Theorem 4.2. Let us start with a property of resistant groups:

Proposition 4.4. Let P be a p-group and B a finite abelian p-group. If P is not resistant then the direct product P × B is not resistant.

Proof : Let G be a finite group with P as Sylow p-subgroup and let Q be an essential p-subgroup of G embedded in P. Such a G exists because we suppose that P is not resistant. In this case P̃ := P × B is a Sylow p-subgroup of G̃ := G × B. As Q is p-centric in P so is Q̃ := Q × B in P̃. Moreover NG̃(Q̃)/Q̃CG̃(Q̃) ≃ N_{G}(Q)/QC_{G}(Q). This means that, as S_{p}(N_{G}(Q)/QC_{G}(Q)) is disconnected, so is S_{p}(NG̃(Q̃)/Q̃CG̃(Q̃)). Then Q̃ is an essential p-subgroup of G̃. This proves that P̃ is not resistant.

Proposition 4.5. Let P = E × A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or is of order p^{3} and exponent p (when p is odd). Then P is not resistant.

Proof : We can realise E as a Sylow p-subgroup of GL(3, F_{p}). One can verify

that Q_{1} = \{( \begin{matrix}1 & 0 & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} )\} and Q_{2} = \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )\} are essential subgroups

of G. So E is not resistant. As P is isomorphic to E × A where A is elementary abelian, by Proposition 4.4, P is not resistant.

In a very recent paper [Pu2], Puig introduced the notion of ’full Frobenius system’ which is a category over a finite p-group P whose objects are the subgroups of P and whose morphisms are a set of injective morphisms between the subgroups of P containing the conjugation by the elements of P. The morphisms satisfy some natural axioms which are inspired by the local properties of P when P is a Sylow p-subgroup of a finite group or a defect group of a block in a group algebra. Puig defined in this context the concept of ’essential group’ and proved that, on a full Frobenius system, the analog of Alperin’s Fusion Theorem holds. Full Frobenius systems are the generalisation of the Frobenius category of a group, and of the Brauer and Puig categories of a block.

The theorem in this paper remains true and all the arguments were chosen to remain valid in a full Frobenius system over P. This permits us to generalize the results to Brauer pairs and pointed groups.

5. Acknowledgments

I thank Rhada Kesser and Markus Linckelmann for their precious suggestions. I am endebted to Prof. Lluis Puig for suggesting the fact that we can extend the main theorem to full Frobenius systems. I am deeply greatefull to Prof. Jacques Thevenaz for his mathematical and moral support all along of the work on this article.

References

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[GM] D. J. Green and P. A. Minh. Almost all Extraspecial p-groups are Swan Groups. Bull. Austral. Math. Soc. 62 (2000), no.1, pp. 149–154.

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[Pu1] L. Puig, Structure locale dans les groupes finis. Bul. Soc. Math. France, Mémoire no. 47, PH thesis, 1976.

[Pu2] L. Puig, Full Frobenius systems and their localizing categories. Preprint, June 2001.

[Th] J. Thévenaz, G-Algebras and Modular Representation Theory. Oxford Science Publications, 1995.

   E-mail address: Radu.Stancu@math.unige.ch

Section de Mathématiques, Université de Genève, CP 240, CH-1211 Genève 24