The maximal number of orbits of a permutation group with bounded movement
type: Item
instance of: scholarly article
title: The maximal number of orbits of a permutation group with bounded movement
zbMATH Open document ID: 0932.20003
author: Pan Soo Kim
author: Cheryl E. Praeger
author: Jung R. Cho
published in: Journal of Algebra
publication date: 1999-12-14T00:00:00Z
review text: If \(G\) is a permutation group on \(\Omega\) with no fixed points, then \(G\) has bounded movement if there is a positive integer \(m\) such that for every \(g\in G\) and \(\Gamma\subseteq\Omega\) the cardinality \(|\Gamma^g\setminus\Gamma|\) is at most \(m\). If \(G\) has bounded movement, then the movement of \(G\) is the least such \(m\). It was shown by \textit{C. E. Praeger} that if \(G\) has movement \(m\) and no fixed points, then the number of \(G\)-orbits is at most \(2m-1\) [see J. Algebra 144, No. 2, 436-442 (1991; Zbl 0744.20004)]. In the present paper, there is a classification of the examples where the bound is attained. Apart from examples with \(|\Omega|=3\), these are elementary Abelian 2-groups of order \(2m\), and the point stabilisers of the \(G\)-orbits are the \(2m-1\) distinct subgroups of index 2. As pointed out by Avinoam Mann, it follows that if \(G\) has movement \(m\), and no fixed points on \(\Omega\), then \(|\Omega|=5m-2\) just when \(G\) is \(S_3\) or \(A_3\) (and \(|\Omega|=3\)). It was shown in the above paper of Praeger that \(|\Omega|\leq 5m-2\). The main theorem follows from another result, due to Peter M. Neumann and proved in this paper. Namely, if \(G\) has \(t\) orbits on \(\Omega\), \(\Gamma\subseteq\Omega\) and the maximum value of \(|\Gamma^g\setminus\Gamma|\) (for \(g\in G\)) is \(m>1\), and \(\Gamma\) has proper non-empty intersection with each \(G\)-orbit, then \(t\leq 2m-1\), and there is a description of the cases when equality holds.
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20B05
zbMATH DE Number: 1295431
zbMATH Keywords: permutation groups
zbMATH Keywords: movement
zbMATH Keywords: numbers of orbits
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
OpenAlex ID: W2024736621
Wikidata QID: Q56988099
cites work: Structure of quasi-invariant sets
cites work: Subsets Close to Invariant Subsets for Group Actions
cites work: Groups Covered By Permutable Subsets
cites work: The structure of finitary permutation groups
cites work: Q3829559
cites work: On permutation groups with bounded movement
DOI: 10.1006/JABR.1998.7704
Recommended article: Q3158485
Recommended article: Intransitive permutation groups with bounded movement having maximum degree
Recommended article: Certain groups with bounded movement having the maximal number of orbits
Recommended article: Q4946063
Recommended article: On the movement of a permutation group
Recommended article: Q5290099
Recommended article: On permutation groups with bounded movement
Recommended article: Improvement on the bounds of permutation groups with bounded movement
Recommended article: Permutation groups with bounded movement having maximum orbits.
Recommended article: Q4217975
description: scientific article; zbMATH DE number 1295431