Groups with small Abelian subgroups
type: Item
instance of: scholarly article
title: Groups with small Abelian subgroups
zbMATH Open document ID: 0637.20013
DOI: 10.1007/BF01187735
author: Avinoam Mann
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: If \(S\) is a subgroup of \(G\) such that \(SZ(G)/Z(G)\) is cyclic, then \(S\) is abelian. The author classifies those finite \(p\)-groups in which all subgroups are of this sort; he shows that in this case \(G/Z(G)\) is either elementary abelian or dihedral or non-abelian of order \(p^3\) and of exponent \(p\). For the two latter cases a detailed description of the isoclinism families is given. Furthermore groups \(G\) with one of the following properties are described: (i) \(G\) has a cyclic Thompson subgroup \(J(G)\), (ii) \(J(G)/Z(G)\) is cyclic, (iii) \(A/Z(G)\) is cyclic for some selfcentralizing normal subgroup \(A\) of \(G\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 4038075
zbMATH Keywords: maximal abelian subgroups
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: isoclinism
zbMATH Keywords: cyclic Thompson subgroup
zbMATH Keywords: selfcentralizing normal subgroup
reviewed by: Hermann Heineken
MaRDI profile type: Publication
cites work: Partitionen endlicher Gruppen
cites work: On groups occurring as center factor groups
cites work: Q4163382
cites work: Gruppen mit kleinen abelschen Untergruppen
cites work: Endliche Gruppen I
description: scientific article; zbMATH DE number 4038075