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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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  • $\begingroup$ By maximal here, you mean that $E$ is elementary abelian of rank $2$, and there are no larger elementary abelian subgroups? $\endgroup$
    – Tobias Kildetoft
    Commented May 20, 2013 at 13:37
  • $\begingroup$ Yes, this is what I mean by maximal. $\endgroup$
    – Jai Mendas
    Commented May 20, 2013 at 19:40
  • $\begingroup$ 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. $\endgroup$
    – Tobias Kildetoft
    Commented May 20, 2013 at 20:01
  • $\begingroup$ Can anyone give a precise reference to the Carlson-Thevenaz paper (I didn't manage to find it in there)? Another question: Isn't $P=\mathbb{Z}/4\times \mathbb{Z}/4$, $E=\mathbb{Z}/2\times \mathbb{Z}/2$ a counterexample? $\endgroup$
    – Kasper Andersen
    Commented Sep 18, 2020 at 13:04

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