Finite self dual groups.
type: Item
instance of: scholarly article
title: Finite self dual groups.
zbMATH Open document ID: 1241.20024
author: Lijian An
author: Jianfang Ding
author: Qinhai Zhang
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
review text: A group \(G\) is said to be \(s\)-self dual if every subgroup of \(G\) is isomorphic to a quotient of \(G\). It is \(q\)-self dual is every quotient of \(G\) is isomorphic to a subgroup of \(G\). It is self dual if it is both \(s\)-self dual and \(q\)-self dual. Since a finite group is self dual \(\iff\) it is nilpotent and all its Sylow subgroups are self dual, it suffices to confine on prime power groups. Every Abelian group is selfdual. A nonabelian \(p\)-group \(G\) is \(s\)-self dual \(\iff\) either \(G=M\times E\), where \(M\) is of order \(p^3\) and exponent \(p\) and \(E\) is elementary Abelian, or \(G=A\times M\), where \(A\) is metacyclic minimal nonabelian of exponent \(p^n>p\) and \(M\) is Abelian of exponent \(<p^n\). The first series of groups and only these groups are self-dual \(p\)-groups.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 6005425
zbMATH Keywords: self dual groups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: minimal non-Abelian \(p\)-groups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.06.014
OpenAlex ID: W1964288093
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DOI: 10.1016/J.JALGEBRA.2011.06.014
description: scientific article; zbMATH DE number 6005425