Cycle types in infinite permutation groups
type: Item
instance of: scholarly article
title: Cycle types in infinite permutation groups
zbMATH Open document ID: 0837.20004
author: Dugald Macpherson
author: Cheryl E. Praeger
published in: Journal of Algebra
publication date: 1996-04-21T00:00:00Z
review text: The main result of the paper is Theorem 1.1: Let \(G\) be a transitive permutation group of an infinite set \(\Omega\) and let \(k\) be a positive integer or \(k = \aleph_0\). Suppose that \(G\) contains an element \(g\) whose cycle decomposition consists of a single infinite cycle, a finite positive number of finite cycles of length at least 2 and at least \(k\) fixed points. (i) If \(k\) is finite, then \(G\) is \((k+1)\)-transitive on \(\Omega\). (ii) If \(k\) is infinite and \((G, \Omega)\) is primitive, then \(G\) is highly transitive. The proof of (ii) is quite hard and makes use of the recent classification of infinite primitive Jordan groups.
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 783920
zbMATH Keywords: infinite primitive permutation groups
zbMATH Keywords: highly transitive groups
zbMATH Keywords: transitive permutation group
zbMATH Keywords: cycle decomposition
zbMATH Keywords: infinite primitive Jordan groups
MaRDI profile type: Publication
full work available at URL: jabr.1995.1184
OpenAlex ID: W2091359338
Wikidata QID: Q56988252
DOI: 10.1006/JABR.1995.1184
description: scientific article; zbMATH DE number 783920