On the largest conjugacy class size in a finite group
type: Item
instance of: scholarly article
title: On the largest conjugacy class size in a finite group
zbMATH Open document ID: 0968.20010
author: John Cossey
author: Trevor Hawkes
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2001-06-10T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108518
review text: Let \(G\) denote a finite group and let \(\text{lcs}(G)\) denote the largest conjugacy class size of \(G\). If \(G_p\) is a Sylow \(p\)-subgroup of \(G\), the authors prove the following Theorem: Let \(G\) be an Abelian-by-nilpotent finite group. Then: \[ \text{lcs}(G)\geq\prod_{p\in\pi(G)}\text{lcs}(G_p). \] The restriction in the above theorem is essential; for every \(\varepsilon>0\) the authors construct an example \(G\) of derived length 3 which satisfies: \[ \text{lcs}(G)<\varepsilon(\prod_{p\in\pi(G)}\text{lcs}(G_p)). \] The proof of the theorem (which goes by induction on the number of prime divisors of \(|G|\)) and the construction of the examples are far from being routine. Taken together, these results illustrate very well the difficulties involved in estimating \(\text{lcs}(G)\).
reviewed by: Marian Deaconescu
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 1544942
zbMATH Keywords: finite groups
zbMATH Keywords: Abelian-by-nilpotent groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: conjugacy class sizes
MaRDI profile type: Publication
cites work: Finite soluble groups
description: scientific article; zbMATH DE number 1544942