SERRE'S THEOREM ON THE COHOMOLOGY ALGEBRA OF A p-GROUP

Abstract

The purpose of this note is to give a proof of a theorem of Serre, which states that if G is a p-group which is not elementary abelian, then there exist an integer m and non-zero elements x₁, … x_m∈H¹ (G, Z/p) such that

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with β the Bockstein homomorphism. Denote by m_G the smallest integer m satisfying the above property. The theorem was originally proved by Serre [5], without any bound on m_G. Later, in [2], Kroll showed that m_G[les ]p^k−1, with k=dim_Z/pH¹ (G, Z/p). Serre, in [6], also showed that m_G[les ](p^k−1)/ (p−1). In [3], using the Evens norm map, Okuyama and Sasaki gave a proof with a slight improvement on Serre's bound; it follows from their proof (see, for example, [1, Theorem 4.7.3]) that m_G[les ](p+1) p^k−2. However, m_G can be sharpened further, as we see below.

For convenience, write H*ast;(G, Z/p)=H*(G). For every x_i∈H¹(G), set

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