A characterization of some finite simple groups by centralizers of elements of order 3

In this article the following theorem is proved.
Theorem. {\it Let $G$ be a finite simple group containing an element $a$ of order $3$ such that $C_G(a)/\langle a\rangle\simeq\operatorname{PSL}(2,q)$, $q >3$. If $C_G(x)$ is a $3$-group for any element $x\in G$ of order $3$ not conjugate with elements in $\langle a\rangle$, then $G$ is isomorphic with one of the groups $M_{23}$, $J_3$ or $\operatorname{PSU}(3,8^2)$}.
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