
\titleofpresentation{Vanishing Products of One Dimensional Classes in
Group Cohomology}

\presenter{Erg\" un Yal\c c\i n}

Let $G$ be a $2$-group. We are interested in the algebra structure
of the cohomology of $G$ in mod $2$ coefficients, denoted usually by
$H^* (G, \mathbb{Z} /2 )$. Let $\sigma _G$ denote the the product of
all (nontrivial) one dimensional classes in $H^* (G, \mathbb{Z}/2)$.
A classical theorem by Serre \cite{yalcine1} states that $(\sigma _G
)^2 =0$ provided that $G$ is not an elementary abelian $2$-group. We
recently improved this result in the following way:

\begin{theorem}[\textnormal{\cite{yalcine2}}] If $G$ is a nonabelian $2$-group, then
$\sigma _G=0$
\end{theorem}

In the proof, we use a cohomological characterization of abelian
$p$-groups. Namely, we show that a $p$-group $G$ is nonabelian if
and only if the inflation map
$$ {\rm inf} ^G _{G/\Phi (G) } : H^3 (G/\Phi (G) , \mathbb{Z} ) \to H^3 (G,
\mathbb{Z})$$ has a nontrivial kernel (here $\Phi (G)$ denotes the
Frattini subgroup of $G$). The key lemma which leads to this
characterization is purely group theoretical.


\begin{ourbiblio}{99}

\bibitem{yalcine1}
J.~P.~Serre, {\em Une relation dans la cohomologie des $p$-groupes},
C.R. Acad. Sci. Paris {\bf 304} (1987), 587--590.

\bibitem{yalcine2}
E.~Yal\c c{\i}n, {\em A note on Serre's theorem in group
cohomology}, preprint 2007, available through author's web page.


\end{ourbiblio}

\presenteraddress{Bilkent University}

\email{yalcine@fen.bilkent.edu.tr}

\web{www.fen.bilkent.edu.tr/~yalcine/}
