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Recognition of the finite simple groups \(S_4(q)\) by their element orders. (Russian, English) Zbl 1067.20016

Algebra Logika 41, No. 2, 166-198 (2002); translation in Algebra Logic 41, No. 2, 93-110 (2002).
It is proved that among the 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 one, are recognizable by the set of their element orders. It is also shown that the 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.

MSC:

20D06 Simple groups: alternating groups and groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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