Products of conjugacy classes of the infinite symmetric groups
type: Item
instance of: scholarly article
title: Products of conjugacy classes of the infinite symmetric groups
zbMATH Open document ID: 0539.20003
DOI: 10.1016/0012-365X(83)90070-5
author: Manfred Droste
published in: Discrete Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let \(S_ 0\) be the group of all permutations on a countably infinite set, and let \(p\in S_ 0\) with at least one infinite orbit. Let \(\pi\) be the following property: (\(\pi)\) p has exactly one infinite orbit and for every natural number n at most a finite number of orbits of length n. One of the main results is the following theorem: (a) If \(\pi\) holds then every element in \(S_ 0\backslash(S^ 0_ 0\backslash A_ 0)\) is the product of 2 conjugates of p; (b) if \(\pi\) does \textit{not} hold then every element of \(S_ 0\) is the product of 2 conjugates of p. (In this statement \(S^ 0_ 0\) is the set of permutations in \(S_ 0\) of finite support and \(A_ 0\) is the infinite alternating group in \(S_ 0)\). The other main result involves permutations on sets of arbitrary infinite cardinality, which in the special case of countable cardinality leads to the conjecture: If \(p\in S_ 0\) has infinite support and is not a fixed- point-free involution then every element of \(S_ 0\) is the product of 3 conjugates of p. A remark added in proof states that the author eventually proved that this conjecture is true.
Mathematics Subject Classification ID: 20B30
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3857350
zbMATH Keywords: infinite orbit
zbMATH Keywords: number of orbits
zbMATH Keywords: product of 2 conjugates
zbMATH Keywords: infinite alternating group
zbMATH Keywords: involution
reviewed by: Joseph L. Zemmer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3857350