Geodesic
A~characterization of the groups $L_3(2^n)$
Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 861-868.

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This note is concerned with finite groups in which a Sylow two-subgroup $S$ has an elementary Abelian subgroup $E$ of order $2^{2n}$, $n\ge2$, such that $E=A\times Z(S)$, $|A|=2^n$, and $C_S(a)=E$ for any involution $a\in A$. It is proved that a simple group satisfying this condition is isomorphic to $L_3(2^n)$. 
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     author = {A. P. Il'inykh},
     title = {A~characterization of the groups $L_3(2^n)$},
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A. P. Il'inykh. A~characterization of the groups $L_3(2^n)$. Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 861-868. http://geodesic.mathdoc.fr/item/MZM_1975_18_6_a7/