A note on Serre ' s theorem in group cohomology

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 H1 (G,Fp). For p = 2, this gives that (σG) = 0, where σG is the product of all nontrivial one dimensional classes in H1 (G, F 2). In this note, we prove that if G is a nonabelian 2-group, then σG is also zero. © 2008 American Mathematical Society.