THE ACTION OF AUTOMORPHISMS ON THE COHOMOLOGY OF A p-GROUP

PETER SYMONDS

Abstract. The automorphism group of a p-group acts naturally on the modp cohomology of the group. This action need not be faithful, but we show that it contains every simple module as a composition factor.

Theorem. Let P be a p-group, for some prime p, and k a field of characteristic p. Then the action of Aut(P) on H^{∗}(P, k) contains every simple kP -module as a composition factor

This was proved by Harris and Kuhn [5] as a consequence of the (proved) Segal Conjecture, but it seems desirable to have an algebraic proof of such a simple result. The author has been shown one by Alperin and Benson [2] for p ≠ 2 which uses varieties. Here we sketch an elementary proof using spectral sequences.

Proof. We can assume that k = 𝔽_{p}. First note that if P is elementary abelian then H^{∗}(P, k) contains the symmetric algebra S^{∗}(βH^{1}(P, k) ≅ S^{∗}(Hom(P, k)), (where β is the Bockstein homomorphism). Now Aut(P) acts faithfully on Hom(P, k), and it follows that the symmetric algebra contains every simple: in fact it contains a free submodule. This can be seen as follows [1] . Let Q be the quotient field of S = S^{∗}(Hom(P, k)). Then Aut(P) acts by automorphisms on Q, and by the Normal Basis Theorem this action is free over the invariant subfield and hence over k. Let U be a free kG-module of rank 1 in Q with generator u, where u = s/t, for s, t ∈ S. Let v be the product of the elements of the orbit of t under Aut(P). Then v is invariant and vU is a free submodule of S.

For the general case we use induction on the order of P. Let Z be the maximal elementary abelian p-subgroup of the center of P. We may assume that P ≠ Z by the discussion above. Consider the natural homomorphism Aut(P) → Aut(Z) × Aut(P/Z). Its kernel is a p-group by [4] 5.3.3., so will not affect the simples. (In fact one can directly identify this kernel with H^{1}(P/Z, Z).)

Consider the terms of the spectral sequence E_{2}^{p,q} = H^{p}(P/Z, H^{q}(Z, k)) ⟹ H^{p+q}(P, k) as Aut(P)-modules. Since Z is central, E_{2}^{p,q} ≅ H^{p}(P/Z, k)⊗_{k} H^{q}(Z, k). According to [3] 10.3.5 and its proof, we can let x_{1}, …, x_{d} be a basis for H^{1}(Z, k) as a k-vector space and let ζ_{i} = N_{Z}^{P} (βx_{i}), (where N denotes the norm, or multiplicative transfer).Then both H^{∗}(P, k) and E_{∞}^{∗,∗} are free modules over the polynomial subring R = k[ζ_{1}, …, ζ_{d}], and also res_{Z}^{P} ζ_{i} = (βx_{i})^{|P/Z|}. Now the module spanned by the (βx_{i})^{|P/Z|} is isomorphic to H^{1}(Z, k) by the Frobenius map x ↦ x^{|P/Z|} (since we assumed that k = 𝔽_{p}), so we obtain a submodule F of E_{∞}^{0,∗} which is free for the image of Aut(P) in Aut(Z).

Let N = Ker(Aut(P) → Aut(Z)). Now Ker(N → Aut(P/Z)) is a p-group, by an earlier remark, so, by induction, H^{∗}(P/Z, k) contains every simple for N. Pick such a simple V and let n be the lowest degree in which it occurs as a composition factor of H^{n}(P/Z, k). No differential d_{r} : E_{r}^{n−r,r} → E_{r}^{n,0} can have image involving V, since its domain is a subquotient of H^{n−r}(P/Z, k) ⊗_{k} H^{r}(Z, k) so does not involve V. Thus E_{∞}^{n,0} contains a simple factor W as an Aut(P)-module such that Res_{N}^{Aut(P)} W contains V.

We claim that the product F W of this with the free Aut(P)/N-module F constructed above is isomorphic to F ⊗_{k} W. Recall that E_{∞}^{∗,∗} is free over R and let I denote the ideal in R of elements of positive degree. Then a basis for H^{∗}(P)/IH^{∗}(P) over k lifts to a basis for H^{∗}(P) over R. But W injects into E_{∞}^{∗,∗}/IE_{∞}^{∗,∗} so the claim is true, and H^{∗}(P, k) has a subquotient isomorphic to F ⊗_{k} W.

Now (k(Aut(P)/N)) ⊗_{k} W ≅ Ind_{N}^{Aut(P)} Res_{N}^{Aut(P)} W, so it involves Ind_{N}^{Aut(P)} V. These, for every V, are enough to generate every simple for Aut(P), because the free module involves every simple yet is induced from N.

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Remark. This proof allows us to calculate explicitly an integer N depending only on |P | such that every simple module occurs in degree less than N. (The existence of such an N is clear.)

Remark. Clearly the action of Aut(P) on H^{∗}(P, k) can have a kernel: consider, for example, the cyclic group of order p^{2}. It seems to be an open question whether the action on H^{∗}(P, ℤ) is always faithful.

References

[1] Alperin, J. Local Representation Theory, Cambridge Studies in Advanced Mathematics, 11, Cambridge University Press, Cambridge, 1986.

[2] Alperin, J. private communication.

[3] Evens, L. The Cohomology of Groups, Oxford University Press, Oxford 1991.

[4] Gorenstein, D. Finite Groups, Harper and Row, New York 1968.

[5] Harris, J. C. and Kuhn, N. Stable decompositions of the classifying spaces of finite abelian p-groups, Math. Proc. Cam. Phil. Soc. 103 427-449, 1988.

Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

   E-mail address: symonds@ms.uky.edu