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For any element $x$ of a group $G$, $x$ is centralized by any element of the subgroup $\langle x \rangle$ generated by $x$.

Is there a name for, or are there any equivalent descriptions for elements $x$ such that $C_G(x) = \langle x \rangle$?

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An element $x\in G$ is said to be "self-centralizing" if $C_G(x) = \langle x\rangle$ (i.e., $x$ commutes only with its powers). See for example Finite simple groups containing a self-centralizing element of order $6$ by John L. Hayden and David L. Winter (Proc. AMS 66 no. 2 (1977), 202-204.

More generally, a subgroup $H$ of $G$ is said to be "self-centralizing" if $C_G(H)=H$.

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    Thanks, that was instructive.2011-05-10