On the order of \(p\)-groups of abundance zero
type: Item
instance of: scholarly article
title: On the order of \(p\)-groups of abundance zero
zbMATH Open document ID: 0932.20013
author: Gustavo A. Fernández-Alcober
author: Raymond T. Shepherd
published in: Journal of Algebra
publication date: 1998-11-29T00:00:00Z
review text: Let \(G\) be a \(p\)-group and \(r(G)\) be the number of conjugacy classes in \(G\). P.~Hall proved that if \(|G|=p^{2n+e}\) with \(e\in\{0,1\}\) then \(r(G)=n(p^2-1)+p^e+l(p^2-1)(p-1)\) for some \(l\geq 0\). Then \(l=l(G)\) is the abundance of \(G\). It is proved that if \(p\geq 11\) is a prime, then there exist no \(p\)-groups of order \(p^{p+2}\) and abundance zero.
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 1214107
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: abundance
zbMATH Keywords: finite \(p\)-groups
reviewed by: N. M. Kurnosenko
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: jabr.1997.7234
OpenAlex ID: W2067109356
cites work: On a special class of \(p\)-groups
cites work: Endliche Gruppen I
cites work: Two problems on finite groups with k conjugate classes
cites work: On \(p\)-groups of maximal class. II
DOI: 10.1006/JABR.1997.7234
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description: scientific article; zbMATH DE number 1214107