Document Zbl 0802.20015

A characterization of some alternating and symmetric groups. (English) Zbl 0802.20015

Commun. Algebra 22, No. 5, 1507-1530 (1994).

Let \(G\) be a finite group and \(\pi_ e(G)\) denotes the set of all orders of elements of \(G\). For a set \(m\) of positive integers let \(h(m)\) be the number of isomorphism classes of finite groups \(G\) such that \(\pi_ e(G) = m\). In this paper the authors prove that \(h(\pi_ e(A_ j)) = 1\) for \(j = 9,11,13\), \(h(\pi_ e(S_ 7)) = 1\) and \(h(\pi_ e(S_ 8)) = \infty\). Earlier it was shown that \(h(\pi_ e(A_ j)) = 1\) for \(j = 5, 7, 8\), \(h(\pi_ e(A_ i)) = \infty\) for \(i = 3, 4, 6\) and \(h(\pi_ e(S_ i)) = \infty\) for \(2 \leq i \leq 6\).

Reviewer: A.Kondrat’ev (Sverdlovsk)

Cited in 4 Reviews

Cited in 19 Documents

MSC:

20D06	Simple groups: alternating groups and groups of Lie type
20D60	Arithmetic and combinatorial problems involving abstract finite groups
20B30	Symmetric groups

Keywords:

alternating and symmetric groups; finite group; orders of elements; isomorphism classes of finite groups

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References:

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