A note on Serre's theorem in group cohomology

Abstract

In 1987, Serre proved that if G is a p-group which is not elementary abelian, then a product of Bocksteins of one dimensional classes is zero in the mod p cohomology algebra of G, provided that the product includes at least one nontrivial class from each line in H 1 (G, Fp). For p = 2, this gives that (σ G ) 2 = 0, where σ G is the product of all nontrivial one dimensional classes in H 1 (G, F 2 ). In this note, we prove that if G is a nonabelian 2-group, then σ G is also zero.