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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?

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By maximal here, you mean that $E$ is elementary abelian of rank $2$, and there are no larger elementary abelian subgroups? – Tobias Kildetoft May 20 '13 at 13:37
Yes, this is what I mean by maximal. – Jai Mendas May 20 '13 at 19:40
Do you still want the subgroup to be maximal (with respect to being elementary abelian)? Otherwise, You will certainly not get them to have cyclic quotients. – Tobias Kildetoft May 20 '13 at 20:01

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