\(p\)-groups of small breadth
type: Item
instance of: scholarly article
title: \(p\)-groups of small breadth
zbMATH Open document ID: 0921.20018
author: Gemma Parmeggiani
author: B. Stellmacher
published in: Journal of Algebra
publication date: 1999-10-03T00:00:00Z
review text: The breadth \(b(G)\) of a finite \(p\)-group \(G\) is given by \(b(G)=\max_{x\in G}b(x)\), where \(p^{b(x)}=|G:C_G(x)|\). It is well-known that \(b(G)\leq 1\) is equivalent to \(|G'|\leq p\). A characterization of groups \(G\) satisfying \(b(G)=2\) and \(b(G)=3\) has been given just recently by \textit{N. Gavioli} et al. [J. Algebra 202, No. 1, 129-141 (1998; see the preceding review Zbl 0921.20017)] for odd \(p\) and \(p\geq 5\), respectively, using character theory. The authors investigate also just these groups but by purely elementary methods. At the same time they close the gaps except \(b(G)=3\) and \(p=2\), where only necessary conditions are derived. The case \(b(G)=3\) and \(p\) odd, for example, is equivalent to: (a) \(|G'|=p^3\) and \(|G:Z(G)|\geq p^4\), or (b) \(|G:Z(G)|=p^4\) and \(|G'|\geq p^4\) or (c) \(|G'|=p^4\) and there exists a normal subgroup \(D\) of \(G\) with \(|D|=p\) and \(|G/D:Z(G/D)|=p^3\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1270873
zbMATH Keywords: centers
zbMATH Keywords: commutator subgroups
zbMATH Keywords: breadth
zbMATH Keywords: finite \(p\)-groups
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
full work available at URL: jabr.1998.7660
OpenAlex ID: W1969078704
cites work: Groups of prime power order with many conjugacy classes
cites work: Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen
cites work: Über den Frobeniusschen Klassenbegriff in nilpotenten Gruppen. II
DOI: 10.1006/JABR.1998.7660
description: scientific article; zbMATH DE number 1270873