Associated primes in group cohomology and a strange result

Abstract

In 2010. Okuyama proved the following. Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p$ dividing the order of $G$. (1) Let $\frak p$ be an associated prime of $H^*(G,k)$ of $\dim=s$. Then there exists an elementary abelian $p$-subgroup $E\subseteq G$ of rank $s$ with $\mathfrak p=\mathfrak p_E$ and depth$\,H^*(C_GE,k)=s$. (2) Let $E$ be an elementary abelian $p$-subgroup of $G$ of rank $s$ and suppose depth$\,H^*(C_GE,k)=s$. Then $\mathfrak p_E$ is an associated prime of $H^*(G,k)$. It is shown this result is true for all fields of characteristic $p$. We also show the following result: If $E$ is an elementary abelian $p$-subgroup of $G$ and depth $H^*(G,k)=\text{rank} E$, then up to conjugation $E$ contains the maximal elementary abelian subgroup contained in the center of a Sylow $p$-subgroup of $G$.