The embedding of a cyclic permutable subgroup in a finite group.
type: Item
instance of: scholarly article
title: The embedding of a cyclic permutable subgroup in a finite group.
zbMATH Open document ID: 1033.20020
author: John Cossey
author: Stewart Stonehewer
published in: Illinois Journal of Mathematics
publication date: 2003-10-16T00:00:00Z
review text: A subgroup \(A\) of a group \(G\) is said to be permutable if \(AX=XA\) for all subgroups \(X\) of \(G\). In an ealier paper [J. Aust. Math. Soc. 71, No. 2, 169-176 (2001; Zbl 1001.20010)], the authors proved that if \(A\) is a cyclic permutable subgroup of odd order of a finite group \(G\), then (1) \([A,G]\) is Abelian; (2) \(A\) acts by conjugation on \([A,G]\) as a group of power automorphisms. The situation turns out to be much more complicated when \(A\) has even order. Results are obtained under the additional condition (*): each cyclic subgroup \(X\) of \(G\) is permutable in \(A_2X\), where \(A_2\) is the \(2\)-component of \(A\). Theorem A: Let \(A\) be a cyclic permutable subgroup of a finite group \(G\) satisfying (*), and let \(N\) be the derived subgroup of \([A,G]\). Then (i) \(N\) has order at most \(2\) and lies in \(A\); and (ii) \(A\) acts on \([A,G]/N\) as a group of universal power automorphisms [a power automorphism is universal if it maps all elements to the same power]. Theorem B: Let \(A\) be a cyclic permutable subgroup of a finite group \(G\) satisfying (*). Then \([A,G]\) is Abelian if and only if \(A\) acts on \([A,G]\) as a group of power automorphisms.
reviewed by: Hans Lausch
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 1993621
zbMATH Keywords: cyclic permutable subgroups
zbMATH Keywords: finite groups
zbMATH Keywords: groups of power automorphisms
MaRDI profile type: Publication
Recommended article: THE EMBEDDING OF A CYCLIC PERMUTABLE SUBGROUP IN A FINITE GROUP. II
Recommended article: Cyclic permutable subgroups of finite groups
Recommended article: Graphical cyclic permutation groups
Recommended article: GROUPS WITH CONJUGATE-PERMUTABLE CONDITIONS
Recommended article: On groups of odd prime-power degree that contain a full cycle
Recommended article: Q4606882
Recommended article: Abelian groups with many automorphisms
Recommended article: Digraph representations of 2-closed permutation groups with a normal regular cyclic subgroup
Recommended article: Conjugate-permutable subgroups
Recommended article: Q5260224
description: scientific article; zbMATH DE number 1993621