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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 617.20045

**Autor: ** Beasley, L.B.; Brenner, J.L.; Erdös, Paul; Szalay, M.; Williamson, A.G.

**Title: ** Generation of alternating groups by pairs of conjugates. (In English)

**Source: ** Period. Math. Hung. 18, 259-269 (1987).

**Review: ** Let A_{n} denote the alternating group of degree n. The main result of the paper is the following Theorem 3.05. Almost all conjugacy classes of A_{n} contain a pair of generators. (In other words, the proportion of conjugacy classes in A_{n} that contain a pair of generators approaches 1 as n ––> oo.)

The main theorem required the proof of the following Theorems 2.04 and 3.04. Let C be a conjugacy class (in the symmetric group of degree n) of type T = 1^{e(1)}2^{e(2)}3^{e(3)}... . If T is not the type of an involution, and if the relation **sum** _{j \geq 1}e(j) \leq n/2 holds, then C contains a pair of elements that generate a primitive group. Almost all partitions of n have a summand > 1 and relatively prime to the other summands.

**Reviewer: ** L.B.Beasley

**Classif.: ** * 20P05 Probability methods in group theory

20F05 Presentations of groups

11P81 Elementary theory of partitions

20D06 Simple groups: alternating and classical finite groups

20D60 Arithmetic and combinatorial problems on finite groups

20B35 Subgroups of symmetric groups

11N45 Asymptotic results on counting functions for other structures

**Keywords: ** alternating group; conjugacy classes; pair of generators; partitions

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