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Recognition of finite groups by a set of orders of their elements

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Abstract

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGLn(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S4(7) is recognizable, whereas A10, U3(5), U3(7), U4(2), and U5(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A5, A7, A8, A9, A11, A12, L2(q), q=7, 8, 11, 49, L3(4), S4(7), U4(3), U6(2), M11, M12, M22, HS, or McL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A6, A10, U3(3), U4(2), U5(2), U3(5), or J2, and G is nonrecognizable; (iii) G is isomorphic to S6(2) or O +8 (2), and h(ω(G))=2.

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  1. V. D. Mazurov
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Supported by RFFR grant No. 96-01-01893.

Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.

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Mazurov, V.D. Recognition of finite groups by a set of orders of their elements. Algebr Logic 37, 371–379 (1998). https://doi.org/10.1007/BF02671691

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  • DOI: https://doi.org/10.1007/BF02671691

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