Evens norm and restriction maps in mod-p cohomology of p-groups

Abstract

Let p be a prime number. Given a p-group G, denote by H*(G) the mod-p cohomology algebra of G. For every subgroup K of G, let Res^G_K be the restriction map from H*(G) to H*(K). Consider the case where K = Ker u, with 0 ≠ u ∈ H¹ (G) = Hom(G,[ ]_p). It was known that, for p = 2, Ker Res^G_K is the principal ideal (u) generated by u. However, for p > 2, a corresponding statement does not hold: while an argument used by Quillen-Venkov in [14] shows that (Ker Res^G_K)² ⊂ (βu) (with β the Bockstein homomorphism), it turns out that, in general, (u, βu) is strictly smaller that Ker Res^G_K; an example is the case where G is any extraspecial p-group of order p³ and K is maximal elementary abelian in G (see [5, 8]).