Title: A note on Serre's theorem in group cohomology

Author: Erg\"un Yal\c c\i n

Address: Department of Mathematics, Bilkent University,
06800 Bilkent, Ankara, Turkey

Current Address (until August 2008): McMaster University,
Department of Mathematics and Statistics,
Hamilton, ON, Canada.


Status of the paper: To appear in Proc. Amer. Math. Soc.


Abstract:
In \cite{Serre2}, Serre proves 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 $H^1 (G, \FF _p)$. For $p=2$, this gives that
$(\sigma _G )^2 =0$ where $\sigma _G$ is the product of all
nontrivial one dimensional classes in $H^1 (G, \FF _2)$. In this
note, we prove that if $G$ is a nonabelian $2$-group, then $\sigma
_G $ is also zero.