Geodesic
A~characterization of some finite simple groups by centralizers of elements of order~3
Sbornik. Mathematics, Tome 37 (1980) no. 4, pp. 489-507

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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 $\operatorname{PSU}(3,8^2)$}. Bibliography: 18 titles. 
@article{SM_1980_37_4_a1,
     author = {B. K. Durakov},
     title = {A~characterization of some finite simple groups by centralizers of elements of order~3},
     journal = {Sbornik. Mathematics},
     pages = {489--507},
     publisher = {mathdoc},
     volume = {37},
     number = {4},
     year = {1980},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1980_37_4_a1/}
}
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B. K. Durakov. A~characterization of some finite simple groups by centralizers of elements of order~3. Sbornik. Mathematics, Tome 37 (1980) no. 4, pp. 489-507. http://geodesic.mathdoc.fr/item/SM_1980_37_4_a1/