Geodesic
Recognition of Finite Simple Groups~$S_4(q)$ by Their Element Orders
Algebra i logika, Tome 41 (2002) no. 2, pp. 166-198.

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It is proved that among simple groups $S_4(q)$ in the class of finite groups, only the groups $S_4(3^n)$, where $n$ is an odd number greater than unity, are recognizable by a set of their element orders. It is also shown that simple groups $U_3(9)$, ${^3D}_4(2)$, $G_2(4)$, $S_6(3)$, $F_4(2)$, and ${^2E}_6(2)$ are recognizable, but $L_3(3)$ is not.
Keywords: finite simple groups, recognizability of groups by their element orders. 
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V. D. Mazurov. Recognition of Finite Simple Groups~$S_4(q)$ by Their Element Orders. Algebra i logika, Tome 41 (2002) no. 2, pp. 166-198. http://geodesic.mathdoc.fr/item/AL_2002_41_2_a4/

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