Lengths of conjugacy classes of finite solvable groups. II
type: Item
instance of: scholarly article
title: Lengths of conjugacy classes of finite solvable groups. II
zbMATH Open document ID: 0798.20014
author: Pamela A. Ferguson
published in: Journal of Algebra
publication date: 1993-03-29T00:00:00Z
review text: [This review concerns also the preceding item Zbl 0798.20013.] Denote by \(w(n)\) the number of distinct prime divisors of the positive integer \(n\); if \(G\) is a finite group, set \(\alpha(G):=\max\{w(| G:C_ G(x)|)\mid x\in G\}\) and \(\beta(G):=\{p\in\pi(G)\mid p|| G:C_ G(x)|\) for some \(x\in G\}\). The most important general result of the papers is that if \(G\) is a solvable group, then \(|\beta(G)|\leq 4\alpha (G)+6\). Several other better estimates are obtained if additional constraints are imposed on \(G\); for example, if \((| G|,6)=1\), then \(|\beta(G)|\leq 3.25\alpha(G) +6\). If \(G/R(G)\) is cyclic, where \(R(G)\) is the intersection of the maximal normal subgroups of \(G\), the former estimates can be improved further to \(|\beta(G)|\leq 3\alpha(G)+6\), or \(|\beta(G)|\leq 3\alpha(G)\) if, moreover, \((| G|,6)=1\).
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D25
zbMATH DE Number: 130514
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Fitting subgroup
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: prime divisors
zbMATH Keywords: finite group
zbMATH Keywords: solvable group
zbMATH Keywords: maximal normal subgroups
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
full work available at URL: jabr.1993.1013
OpenAlex ID: W4213377440
DOI: 10.1006/JABR.1993.1013
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description: scientific article; zbMATH DE number 130514