Conjugacy class lengths of metanilpotent groups
type: Item
instance of: scholarly article
title: Conjugacy class lengths of metanilpotent groups
zbMATH Open document ID: 0879.20011
author: Carlo Casolo
author: Silvio Dolfi
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1997-08-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108403
review text: For a finite group \(G\), let \(\pi(G)\) denote the set of prime divisors of \(|G|\); if \(g\in G\), let \(\sigma(g)\) be the set of prime divisors of \(|G:C_G(g)|\), let \(\sigma^*(G)=\max\{|\sigma(g)|:g\in G\}\) and let \(\rho^*(G)=\pi(G/Z(G))\). B. Huppert has asked whether the inequality \(|\rho^*(G)|\leq 2\sigma^*(G)\) holds for soluble groups \(G\). The authors prove that if \(G\) is a finite metanilpotent group and if \(p\) is the smallest prime divisor of \(|G:G'Z(G)|\), then \(|\rho^*(G)|<((2p-1)/(p-1))\sigma^*(G)\). This inequality is then shown to be best possible by letting \(p=2\) and constructing an infinite family of supersoluble metabelian groups \(G_n\) satisfying \(\lim_{n\to\infty}|\rho^*(G_n)|/\sigma^*(G_n)=3\). Thus Huppert's question is solved in the negative, in spite of a number of particular cases in which it is known to hold (it was verified for \(\sigma^*(G)=1\) by \textit{D. Chillag} and \textit{M. Herzog} [J. Algebra 131, No. 1, 110-125 (1990; Zbl 0694.20015)], for \(\sigma^*(G)=2\) by \textit{P. A. Ferguson} [J. Algebra 143, No. 1, 25-28 (1991; Zbl 0735.20011)] and for \(\sigma^*(G)=3\) by \textit{C. Casolo} [Manuscr. Math. 82, No. 2, 171-189 (1994; Zbl 0819.20024)]).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 11N45
zbMATH DE Number: 1021765
zbMATH Keywords: conjugacy classes
zbMATH Keywords: finite groups
zbMATH Keywords: soluble groups
zbMATH Keywords: finite metanilpotent groups
zbMATH Keywords: supersoluble metabelian groups
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1021765