Regular orbits of cyclic subgroups in permutation representations of certain simple groups
type: Item
instance of: scholarly article
title: Regular orbits of cyclic subgroups in permutation representations of certain simple groups
zbMATH Open document ID: 1028.20010
DOI: 10.1016/S0021-8693(02)00107-2
author: Johannes Siemons
author: A. Zalesski
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: Let \(G\) be a group and \(\Omega\) a set. If \(G\) acts on \(\Omega\), then a \(G\)-orbit is regular if its cardinality is \(|G|\). The authors study regular orbits of cyclic subgroups of finite simple groups. The main theorem of this paper is the following theorem: Theorem 1.1. Let \(G\) be a known finite simple group, not isomorphic to an alternating group \(A_n\), which admits a doubly transitive permutation representation. Then every cyclic subgroup \(H\subset G \) has a regular orbit in any non-trivial permutation representation of \(G\). The authors [in J. Algebra 226, No. 1, 451-478 (2000; Zbl 0957.20013)] proved this theorem for \(\text{PSL}(n,q)\).
reviewed by: A. Iranmanesh
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C34
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20B20
zbMATH DE Number: 1868050
zbMATH Keywords: finite simple groups
zbMATH Keywords: regular orbits
zbMATH Keywords: cyclic subgroups
zbMATH Keywords: doubly transitive groups
zbMATH Keywords: permutation representations
MaRDI profile type: Publication
cites work: On the maximal subgroups of the finite classical groups
cites work: Q3684278
cites work: Q4882944
cites work: Singer-Zyklen in klassischen Gruppen
cites work: Endliche Gruppen I
cites work: Q3996618
cites work: The maximal factorizations of the finite simple groups and their automorphism groups
cites work: Intersections of matrix algebras and permutation representation of \(\text{PSL}(n,q)\)
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description: scientific article; zbMATH DE number 1868050