A note on barely transitive permutation groups satisfying min-\(2\)
type: Item
instance of: scholarly article
title: A note on barely transitive permutation groups satisfying min-\(2\)
zbMATH Open document ID: 0799.20003
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1994-03-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108312
review text: A permutation group \(G\) on an infinite set \(\Omega\) is barely transitive if it is transitive but for every proper subgroup \(K\) of \(G\), all the \(K\)-orbits on \(\Omega\) are finite. In this paper the author investigates locally finite barely transitive groups \(G\) with point stabilizer \(H\). It is shown that if \(G\) has an involution and \(H\) satisfies min-2, then \(G\cong C_{2^ \infty}\). (This was proved by the author in an earlier paper with any prime \(p\) in place of 2, but using the classification of finite simple groups; the present proof does not, but uses the Feit- Thompson Theorem). Other results in the paper include the following: (1) If for some prime \(p\), \(H\) has min-\(p\), then so does \(G\). (2) If for some prime \(p\), every \(p\)-subgroup of \(H\) is soluble, then \(G\) is a union of a chain of proper normal subgroups. (3) If the order of every simple section of \(H\) is bounded, then the conclusion of (2) holds. (4) If there are \(Y \triangleleft X < G\) with \(X/Y\) an infinite simple group, then \(Y\) is infinite and not locally soluble, and \(H\) involves an isomorphic copy of \(X/Y\).
Mathematics Subject Classification ID: 20B35
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 20F50
Mathematics Subject Classification ID: 20B22
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 510093
zbMATH Keywords: union of a chain of normal subgroups
zbMATH Keywords: permutation group
zbMATH Keywords: locally finite barely transitive groups
zbMATH Keywords: point stabilizer
zbMATH Keywords: involution
zbMATH Keywords: min-2
zbMATH Keywords: min-\(p\)
zbMATH Keywords: \(p\)-subgroup
zbMATH Keywords: simple section
zbMATH Keywords: infinite simple group
author: Mahmut Kuzucuoğlu
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
cites work: Locally finite groups with Chernikov Sylow p-subgroups
cites work: On the normalizer condition and barely transitive permutation groups
cites work: Q3351597
cites work: Q5675882
cites work: Barely transitive permutation groups
cites work: On the minimality property for locally finite groups
description: scientific article; zbMATH DE number 510093