Geodesic
Groups with a centralizer of sixth order
Matematičeskie zametki, Tome 22 (1977) no. 1, pp. 153-159.

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Let $G$ be a finite fusion-simple group with a self-centralizing subgroup $A$ of sixth order. It is proved that if the centralizer of the involution from $A$ is an unsolvable subgroup of $G$ of an odd index, then $G$ is isomorphic with the Janko group $J_1$. 
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     author = {A. A. Makhnev},
     title = {Groups with a centralizer of sixth order},
     journal = {Matemati\v{c}eskie zametki},
     pages = {153--159},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_1_a17/}
}
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A. A. Makhnev. Groups with a centralizer of sixth order. Matematičeskie zametki, Tome 22 (1977) no. 1, pp. 153-159. http://geodesic.mathdoc.fr/item/MZM_1977_22_1_a17/