Groups of prime power order with many conjugacy classes
type: Item
instance of: scholarly article
title: Groups of prime power order with many conjugacy classes
zbMATH Open document ID: 0921.20017
author: Avinoam Mann
author: Valerio Monti
author: Andrea Previtali
author: Norberto Gavioli
author: Carlo Maria Scoppola
published in: Journal of Algebra
publication date: 1999-10-03T00:00:00Z
review text: Concerning the number \(k(G)\) of conjugacy classes of a finite \(p\)-group \(G\) the authors investigate on the one hand the structure of groups with relatively large \(k(G)\), on the other hand they give estimates of values, \(k(G)\) cannot achieve. With respect to the latter question they derive for example the following result: There exists no \(p\)-group \(G\) (\(p>2\)) of order \(p^n\) such that \(3p^{n-3}\leq k(G)\leq p^{n-2}+2p-3\). Regarding the former problem Theorem 1 states, that \(k(G)\geq 3| G|/p^3\) implies (i) \(| G_2|\leq p^2\) or (ii) \(| G:Z(G)|\leq p^3\), or (iii) \(G\) has an abelian maximal subgroup. A similar result is proved if \(k(G)\geq 5| G|/p^4\). Corollaries relate among others to \(p\)-groups of breadth 2 or 3. Using other methods and thereby eliminating some excluded cases the results on the breadth of \(G\) have been proved just recently by \textit{G. Parmeggiani} and \textit{B. Stellmacher} [in J. Algebra 213, No. 1, 52-68 (1999; see the following review Zbl 0921.20018)].
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1203887
zbMATH Keywords: irreducible characters
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: breadth
reviewed by: Gerhard Kowol
MaRDI profile type: Publication
OpenAlex ID: W1970514896
cites work: Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen
cites work: Über den Frobeniusschen Klassenbegriff in nilpotenten Gruppen. II
cites work: Q4103728
cites work: Two Combinatorial Problems in Group Theory
cites work: Q3961021
cites work: Breadth, class and commutator subgroups of p-groups
DOI: 10.1006/JABR.1997.7222
description: scientific article; zbMATH DE number 1203887