Timeline for center of centralizer in finite group

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Dec 23, 2014 at 14:31	comment	added	Hamid		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.
Dec 23, 2014 at 11:52	comment	added	Derek Holt		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 $\|x\|=p$.
Dec 23, 2014 at 10:23	history	asked	Hamid	CC BY-SA 3.0