Let $P$ be a $p$-group. It is known that if $E$ is a maximal elementary abelian subgroup of rank 2 in $P$, then $C_P(E)/E$ is cyclic where $C_P(E)$ denotes the centralizer of $E$ in $P$. This is proved, for example, in the paper by Jon Carlson and Jacques Thevenaz on the endotrivial modules.

I wonder if there is such a general result for any elementary abelian subgroup?