ALMOST ALL EXTRASPECIAL p-GROUPS ARE SWAN GROUPS

DAVID JOHN GREEN AND PHAM ANH MINH

Abstract. Let P be an extraspecial p-group which is neither dihedral of order 8, nor of odd order p^{3} and exponent p. Let G be a finite group having P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer N_{G}(P).

Introduction

Let P be a finite p-group. Martino and Priddy call P a Swan group [5] if for every finite group G with Sylow p-subgroup P, the mod-p cohomology ring H^{∗}(G) coincides with H^{∗}(N_{G}(P)) ≅ H^{∗}(P)^{N_{G}(P)}. In particular, if there are no so-called transfer summands in the stable decomposition of the classifying space BP, then P is a Swan group.

We prove in Theorem 3.3 that all extraspecial p-groups are Swan groups, apart from the well-known exceptions 2_{+}^{1+2} = D_{8} and (for p odd) p_{+}^{1+2} = E. The cases where P is the metacyclic group p_{−}^{1+2} = M(p^{3}) with p odd, and where P is 2_{−}^{1+2n} = Q_{8} ∗ D_{8} ∗ ⋯ ∗ D_{8}, were proved in [5]; the former case being due to G. Glauberman. Earlier, the p = 2 case of the theorem was published in [7], but with an incorrect proof: see Remark 3.6. In Corollary 3.5 we generalize another result of Martino and Priddy, exhibiting three infinite families of Swan groups whose classifying spaces do have transfer summands in their stable decompositions.

Throughout this paper we denote the mod-p cohomology ring H^{∗}(G, 𝔽_{p}) by H^{∗}(G). A suitable reference on group cohomology is Evens’ book [2].

1. Extraspecial p-groups

Recall that a p-group P is called extraspecial if its centre Z(P), its derived subgroup P′ and its Frattini subgroup Φ(P) all coincide, and are cyclic of order p. So if P is extraspecial there is a central extension

1 ⟶ 𝔽_{p} ⟶ P ⟶^{ψ} V ⟶ 1

with V an elementary abelian p-group. Hence there is a nondegenerate alternate bilinear form f on V defined by f(ψ(g), ψ(h)) = [g, h] for all g, h ∈ P.

Moreover, for p odd there is a linear form λ on V defined by λ(ψg) = g^{p}; and for p = 2 there is a quadratic form Q on V defined by Q(ψg) = g^{2} with associated bilinear form f. Conversely, such a pair (λ, f) determines an extraspecial p-group when p is odd, and such a Q determines an extraspecial 2-group.

See [4, Ch. 6] for a reference on alternate bilinear forms, and [1, I.16] for a reference on quadratic forms in characteristic 2. Nondegeneracy means that V has even dimension, say 2n. Up to change of basis for V there are two possibilities for Q when p = 2, and exactly one choice of f for odd p. For p odd there are two possibilities for the pair (λ, f): either λ is identically zero, or it is not. Note that in the case of nonzero λ, Witt’s extension theorem does not hold for the pair (λ, f), as the restriction of f to ker(λ) does have a kernel.

Date: 27 October 1999.

1991 Mathematics Subject Classification. Primary 20J06; Secondary 20D15, 55R35.

So there are four types of extraspecial p-groups. In each case P is generated by A_{1}, …, A_{n}, B_{1}, …, B_{n}, C, with C central of order p, [A_{i}, A_{j}] = [B_{i}, B_{j}] = 1 and [A_{i}, B_{j}] = C^{δ_{ij}}. Moreover A_{i}^{p} = B_{i}^{p} = 1 for 2 ≤ i ≤ n. The four cases are:

• 2_{+}^{1+2n} = D_{8} ∗ ⋯ ∗ D_{8}: here A_{1}^{2} = B_{1}^{2} = 1.

• 2_{−}^{1+2n} = Q_{8} ∗ D_{8} ∗ ⋯ ∗ D_{8}: here A_{1}^{2} = B_{1}^{2} = C.

• p_{+}^{1+2n} = E ∗ ⋯ ∗ E has odd exponent p: here A_{1}^{p} = B_{1}^{p} = 1.

• p_{−}^{1+2n} = M(p^{3}) ∗ E ∗ ⋯ ∗ E has odd exponent p^{2}: here A_{1}^{p} = C and B_{1}^{p} = 1.

The characteristic subgroup Ω_{1}(P) of P is the subgroup generated by all order p elements. Denote Z(Ω_{1}(P)) by Y. If P has odd exponent p^{2} then Y = ⟨B_{1}, C⟩ is rank two elementary abelian; in all other cases, Y equals Z = Z(P).

The following result could be called Witt’s theorem for extraspecial p-groups.

Proposition 1.1. Let P be an extraspecial p-group. Suppose that H, K are subgroups of P containing Z, and that ϕ: H → K is a group isomorphism inducing the identity map on Z. If P has odd exponent p^{2}, assume further that H ∩ Y = K ∩ Y and that ϕ induces the identity map on (H ∩ Y)/Z. Then ϕ extends to an automorphism of P.

Lemma 1.2. In Proposition 1.1, suppose that P has odd exponent p^{2} and that H ∩ Y is Z rather than Y. Then ϕ extends to an isomorphism from ⟨H, Y ⟩ to ⟨K, Y ⟩ which itself satisfies the conditions of Proposition 1.1.

Proof. Since ϕ(C) = C it follows that h^{−1}ϕ(h) lies in ker(λ) for every h ∈ H. Hence ϕ([h, B_{1}]) = [ϕ(h), B_{1}]. So we may set ϕ(B_{1}) = B_{1}.

Proof of Proposition 1.1. Denote by U, W the images in V of H, K respectively. Since ϕ is the identity on Z there is an 𝔽_{p}-vector space isomorphism ρ: U → W induced by ϕ which respects the alternate bilinear form f on V.

If p is 2 then ρ respects the quadratic form Q. Since Witt’s extension theorem holds for Q (see [1, p. 36]), we may extend ρ to a Q-orthogonal transformation ρ of V. Using the standard generators for P we may lift ρ to an automorphism ϕ′ of P. If h ∈ H then ϕ′(h) = ϕ(h)C^{r} for some r ∈ ℤ/p. Since P has enough inner automorphisms, we may assume that ϕ′ extends ϕ.

To be more precise: pick h_{1}, …, h_{m} ∈ H whose images under ψ: P → V constitute a basis for U. Since the alternate bilinear form f on V is nondegenerate we can pick g_{1}, …, g_{m} ∈ P such that f(ψ(g_{i}), ψ(ϕh_{j})) = δ_{ij}. Hence conjugation by g_{i} fixes ϕ(h_{j}) for j ≠ i and sends ϕ(h_{i}) to ϕ(h_{i})C. So we can correct ϕ′ by an inner automorphism of P to ensure that ϕ′ = ϕ on H.

Now suppose that p is odd. As Witt’s extension theorem holds for f (see [4, 6.9]), we may extend ρ to a transformation of V that respects f. But then ρ respects λ too: this is trivial in the exponent p case, as λ is then zero. In the exponent p^{2} case, we may assume by Lemma 1.2 that H, K contain Y and that ρ fixes ψ(B_{1}). So ρ respects λ by Lemma 1.3 below. As in the p = 2 case we can now lift ρ to an automorphism of P which extends ϕ.

Lemma 1.3. Suppose P is an extraspecial p-group with odd exponent p^{2}. Taking C as basis for Z ≅ 𝔽_{p}, we may assume that λ(ψA_{1}) = 1. Then for any v ∈ V we have λ(v) = f(v, ψ(B_{1})).

Proof. Each element g of P has canonical form B_{1}^{s_{1}} ⋯ B_{n}^{s_{n}} · A_{1}^{r_{1}} ⋯ A_{n}^{r_{n}} · C^{t}. Then g^{p} = C^{r_{1}} = [g, B_{1}].

2. Local subgroup structure

Throughout this section G is a finite group with extraspecial Sylow p-subgroup P.

Lemma 2.1. Suppose that P is not one of D_{8}, E, M(p^{3}). Then for any order p element g of P, the centralizer of g in P has the same Frattini subgroup as P itself.

Proof. If n ≥ 2 then C_{P} (g) is not abelian. In Q_{8}, all order p elements are central. The other three groups really are exceptions: take g = B_{1}.

Lemma 2.2. Suppose that the centralizer of each exponent p element of P has Z as its Frattini subgroup. Then Z^{g} = Z for every g ∈ G such that P ∩P^{g} contains Z. Moreover, such g factorize as g = g_{1}g_{2} with g_{1} ∈ N_{G}(P) and g_{2} ∈ C_{G}(Z).

Proof. Observe that P is a Sylow p-subgroup of C_{G}(Z), and so all Sylow p-subgroups of C_{G}(Z) have Frattini subgroup Z. Now set R equal to P^{g} ∩ C_{G}(Z). By assumption the Frattini subgroup of R is that of P^{g}, namely Z^{g}. But R is contained in a Sylow p-subgroup of C_{G}(Z). We conclude that Z^{g} = Z.

Therefore P^{g} is itself a Sylow p-subgroup of C_{G}(Z), and so P^{g} = P^{h} for some h ∈ C_{G}(Z). Take g_{1} = gh^{−1} and g_{2} = h.

Lemma 2.3. Suppose that P has odd exponent p^{2}. Then Y^{g} = Y for every g ∈ C_{G}(Z) such that Y ≤ P ∩ P^{g}. Moreover, such g factorize as g_{1}g_{2}, where g_{1} lies in C_{G}(Z) ∩ N_{G}(P) and g_{2} ∈ C_{G}(Z) ∩ N_{G}(Y) acts trivially on Y/Z.

Proof. Set D_{1} equal to C_{G}(Y), which contains Ω_{1}(P). Since the centre of P is cyclic and Ω_{1}(P) is maximal in P, it follows that Ω_{1}(P) is a Sylow p-subgroup of D_{1}. Now let R be P^{g} ∩ D_{1}, the centralizer of Y in P^{g}. Since Ω_{1}(P) has exponent p, so does R. As g centralizes Z, we deduce that R is a maximal subgroup of P^{g}. But the only exponent p maximal subgroup of P^{g} is Ω_{1}(P^{g}), which has centre Y^{g}. So g normalizes Y.

Now set D_{2} = {h ∈ C_{G}(Z) ∩ N_{G}(Y) | g acts trivially on Y/Z}. Then P is a Sylow p-subgroup of D_{2}, and P^{g} is too since g lies in C_{G}(Z) ∩ N_{G}(Y). So P^{g} = P^{h} for some h ∈ D_{2}. Take g_{1} = gh^{−1} and g_{2} = h.

3. Stability conditions

The following elementary reformulation of the usual stability condition is not new, but does not appear to be widely known.

Lemma 3.1. Let P be a Sylow p-subgroup of a finite group G. The cohomology class x ∈ H^{∗}(P, 𝔽_{p}) lies in Im Res_{P}^{G} if and only if x is an N_{G}(P)-invariant and

Cor_{P ∩P^{g}}^{P} Res_{P ∩P^{g}}^{P^{g}} g^{∗}(x) = 0 for all g ∈ G − N_{G}(P).

Proof. If x comes from H^{∗}(G), then it certainly satisfies both conditions. Conversely, observe that the conditions combined with the Mackey formula mean that Res_{P}^{G} Cor_{P}^{G}(x) = |N_{G}(P) : P |x.

Let P be a finite p-group. The ring of universally stable elements I(P) was defined in [3] as the subring of H^{∗}(P, 𝔽_{p}) given by

I(P) = ⋂_{G} Im Res_{P}^{G} ,

where G ranges over all finite groups with Sylow p-subgroup P. The following observation appears in [7]. Recall that O^{p}(G) is the subgroup generated by all p′-elements of G.

Lemma 3.2. I(P) ⊆ H^{∗}(P)^{O^{p}(Out(P))}.

Proof. Pick any outer automorphism of order prime to p, and lift it to an automorphism α of the same order. Let G be the semidirect product P ⋊ ⟨α⟩.

It is immediate that if P is a Swan group then equality holds in Lemma 3.2. The main result of this paper is:

Theorem 3.3. All extraspecial p-groups P apart from D_{8} and E are Swan groups. If P is D_{8} or E, then the universally stable elements for P are strictly contained in H^{∗}(P)^{O^{p}(Out(P))}.

Proof. See [5] for a proof that M(p^{3}) is a Swan group. The groups D_{8} and E are treated in Lemma 3.4 below. So we may assume that P satisfies the hypotheses of Lemma 2.1 and hence those of Lemma 2.2.

Let G be a finite group with extraspecial Sylow p-subgroup P, and let x ∈ H^{∗}(P) be an N_{G}(P)-invariant. We show that the conditions of Lemma 3.1 are satisfied. Let g be an element of G−N_{G}(P). If P^{g} ∩P is elementary abelian but not maximal in P, then corestriction from P^{g} ∩ P to P is zero: for corestriction from any group H to H × C_{p} is zero.

We may therefore assume that P^{g}, P both contain Z = Φ(P). Write H for P ∩P^{g} and L for ^{g}P ∩ P. By Lemma 2.2 we deduce that g normalizes Z. Moreover, since x is invariant under N_{G}(P), we may in fact assume that g stabilizes Z. If P has odd exponent p^{2} we deduce further by Lemma 2.3 either that g normalizes Y and can be taken to act trivially on Y/Z, or that H ∩ Y = L ∩ Y = Z.

So we can now apply Proposition 1.1 and deduce that conjugation c_{g}: H → L extends to an automorphism ϕ of P. Then Res_{H}^{P^{g}} g^{∗} = Res_{H}^{P} ϕ^{∗}, which means that Cor_{H}^{P} Res_{H}^{P^{g}} g^{∗} = 0 as H is a proper subgroup of P.

Lemma 3.4. If P is D_{8} or E, then the inclusion in Lemma 3.2 is strict.

Proof. Let G be GL_{3}(𝔽_{p}). Then the upper triangular matrices with ones on the diagonal form a Sylow p-subgroup isomorphic to P. We may take B_{1} = ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} )

and C = ( \begin{matrix}1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ). Pick g = ( \begin{matrix}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ). Then F = P ∩ P^{g} equals ⟨B_{1}, C⟩, a maximal

elementary abelian subgroup. Let β, γ be the dual basis for F^{∗}.

Taking the first Chern class is an 𝔽_{p}-linear monomorphism from F^{∗} to H^{2}(F). Let ρ be an ordinary representation of P with character χ, the sum of all p^{2} linear characters. These linear characters restrict to F as scalar multiples of β, each scalar multiple being the image of p characters. So by the Whitney sum formula, the total Chern class c(ρ) restricts to F as follows:

Res_{F}^{P} c(ρ) = ∏_{μ∈𝔽_{p}}(1 + μβ)^{p} .

This equals 1 − β^{p(p−1)}. Set η equal to c_{p(p−1)}(ρ) in H^{2p(p−1)}(P). Then η lies in H^{∗}(P)^{O^{p}(Out(P))}, since χ is an invariant of Aut(P). But Res_{F}^{P} (η) = −β^{p(p−1)} is distinct from g^{∗} Res_{F}^{P} (η) = −γ^{p(p−1)}, so η is not stable.

Suppose that P is a p-group whose classifying space does not have any transfer summands in its stable splitting. Then P is a Swan group by Theorem 3.5 of [5]. Martino and Priddy give one counterexample to the converse: M(p^{3}).

Corollary 3.5. Let P be 2_{+}^{1+2n} with n ≥ 2; or p_{+}^{1+2n} with p odd and n ≥ 2; or p_{−}^{1+2n} with p odd and n ≥ 1. Then although P is a Swan group, the stable splitting of BP involves a transfer summand.

Proof. F = ⟨B_{1}, …, B_{n}, C⟩ is a self-centralising, maximal elementary abelian subgroup. By Theorem 0.1 of [8], the Steinberg summand L(n + 1) of BF is a transfer summand of BP.

Remark 3.6. Ogawa’s proof of Theorem 3.3 hinges around the following claim: if M is a maximal subgroup of an extraspecial 2-group P ≠ D_{8}, then P acts trivially on H^{∗}(M). As observed in [6], the proof of this claim in [7] uses inflation incorrectly. We shall now see that the claim is false for 2_{+}^{1+2n}.

Let P be D_{8}∗⋯∗D_{8}, or p_{+}^{1+2n} if p is odd. Let M be C_{P} (B_{1}), an index p subgroup of P. Then F = ⟨B_{1}, …, B_{n}, C⟩ is maximal elementary abelian in M and in P. Let β_{1}, …, β_{n}, γ be the dual basis of F^{∗}, which we embed in H^{2}(F) by taking the first Chern class. Then S(F^{∗}) ⊆ H^{∗}(F). Define ζ ∈ H^{2p^{n−1}}(M) to be the Evens norm N_{F}^{M}(γ), and set g = A_{1}. Let U be the subspace of F^{∗} spanned by β_{2}, …, β_{n}. By standard properties of the Evens norm map (see [2, Ch. 6]), we have:

Res_{F}^{M}(η) = ∏_{u∈U}(γ + u) and so g^{∗} Res_{F}^{M}(η) = ∏_{u∈U}(γ + β_{1} + u) .

These restrict to ⟨B_{1}, C⟩ as γ^{p^{n−1}} and (γ + β_{1})^{p^{n−1}} respectively. So Res_{F}^{M}(g^{∗}η) differs from Res_{F}^{M}(η), which means that η is not invariant under A_{1}.

References

[1] J. A. Dieudonné. La géométrie des groupes classiques. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 5. Springer-Verlag, Berlin, third edition, 1971.

[2] L. Evens. The cohomology of groups. Oxford Univ. Press, Oxford, 1991.

[3] L. Evens and S. Priddy. The ring of universally stable elements. Quart. J. Math. Oxford Ser. (2), 40(160):399–407, 1989.

[4] N. Jacobson. Basic algebra. I. W. H. Freeman and Company, New York, second edition, 1985.

[5] J. Martino and S. Priddy. On the cohomology and homotopy of Swan groups. Math. Z., 225(2):277–288, 1997.

[6] P. A. Minh. Proper singularities of extraspecial p-groups for the mod-p cohomology functor. Comm. Algebra, 25(3):965–971, 1997.

[7] Y. Ogawa. On the subring of universally stable elements in a mod-2 cohomology ring. Tokyo J. Math., 15(1):91–97, 1992.

[8] S. B. Priddy. On characterizing summands in the classifying space of a group. I. Amer. J. Math., 112(5):737–748, 1990.

Dept of Mathematics, Univ. of Wuppertal, D–42097 Wuppertal, Germany

   E-mail address: green@math.uni-wuppertal.de

Dept of Mathematics, University of Hue, 27 Nguyen Hue, Hue, Vietnam