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 An denote the alternating group of degree n. The main result of the paper is the following Theorem 3.05. Almost all conjugacy classes of An contain a pair of generators. (In other words, the proportion of conjugacy classes in An 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 = 1e(1)2e(2)3e(3)... . If T is not the type of an involution, and if the relation sum j \geq 1e(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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