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My general question: Is there any reference for the center of centralizer in finte group. In particular for the element $x\in G$ such that $Z(C_G(x))=\langle x\rangle$.

My motivation: Espacially when $G$ is almost simple ($F^*(G)$ is simple) and $x$ has prime order is iteresting for me.

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    $\begingroup$ In general it could be any abelian group containing$\langle x \rangle$. Even in the specific case you mention I think the question is too vague, and you need to specify the kind of the kind of properties you are looking for. There are certainly examples where $Z(C_G(x))$ strictly contains $\langle x \rangle$, such as $x \in {\rm PSL}(2,p^n)$ for $n>1$ with $|x|=p$. $\endgroup$
    – Derek Holt
    Commented Dec 23, 2014 at 11:52
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    $\begingroup$ Professor Holt@ I think it is well known fact when an almost simple group $L$ with socle $K$ ($K$ is simple Lie type) contains an field or graph automorphism $x$ of prime order, then there exist an element $y\in xK$ with the above property. I hope this makes clear my motivation. $\endgroup$
    – Hamid
    Commented Dec 23, 2014 at 14:31

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