Groups which are isomorphic to their nonabelian subgroups
type: Item
instance of: scholarly article
title: Groups which are isomorphic to their nonabelian subgroups
zbMATH Open document ID: 0887.20012
author: Howard Smith
author: James Wiegold
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1998-03-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108432
review text: Let \(\mathcal X\) be the class of all groups \(G\) containing proper non-abelian subgroups and such that all these subgroups are isomorphic to \(G\). The authors give a complete description of soluble \(\mathcal X\)-groups, and prove that if \(G\) is an unsoluble \(\mathcal X\)-group, then \(G/Z(G)\) is simple and each non-abelian subgroup of \(G\) contains \(Z(G)\). It follows in particular that every locally graded \(\mathcal X\)-group is soluble (a group \(G\) is locally graded if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index). The problem of the existence of unsoluble \(\mathcal X\)-groups is left here as an open question.
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20E25
zbMATH DE Number: 1089765
zbMATH Keywords: nonabelian subgroups
zbMATH Keywords: soluble groups
zbMATH Keywords: locally graded groups
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: groups isomorphic to proper subgroups
MaRDI profile type: Publication
cites work: Groups with restricted non-normal subgroups
cites work: On Finitely Generated Soluble Groups With No Large Wreath Product Sections
cites work: A problem about normal subgroups
cites work: Number theory, Carbondale 1979. Proceedings of the Southern Illinois number theory conference, Carbondale, March 30 and 31, 1979
cites work: Q3994685
cites work: Q3663506
cites work: On homomorphic images of locally graded groups
cites work: Q3813882
description: scientific article; zbMATH DE number 1089765