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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 $$n \geq 9$$.

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