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A new characterization of the alternating groups. (English) Zbl 0790.20030
For a finite group \(G\), let \(\pi_ e(G)\) denote the set of orders of elements of \(G\). Starting with results of G. Zacher [Rend. Semin. Mat. Univ. Padova 27, 267-275 (1957; Zbl 0166.288)], G. Higman [J. Lond. Mat. Soc. 32, 335-342 (1957; Zbl 0079.03204)] and R. Brandl [Boll. Unione Mat. Ital., V. Ser., A 18, 491-493 (1981; Zbl 0473.20013)] about the case where \(\pi_ e(G)\) consists of prime powers, there is a long series of papers dealing with characterizations of \(G\) by \(\pi_ e(G)\).
The present paper is devoted to the following: Conjecture. Let \(G\) be a group and \(M\) a finite simple group. Then \(G\cong M\) if and only if (a) \(\pi_ e(G) = \pi_ e(M)\), and (b) \(| G| = | M|\).
It is shown that the conjecture holds for the alternating groups \(A_ n\). Indeed, for \(n=5\), 7 and 8, the groups \(A_ n\) are already characterized by condition (a). An analogous statement is false for \(n = 6\) but it seems open whether it is true for all \(n \geq 9\).

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