Finite groups with many conjugate elements
type: Item
instance of: scholarly article
title: Finite groups with many conjugate elements
zbMATH Open document ID: 0816.20023
published in: Journal of Algebra
publication date: 1994-12-19T00:00:00Z
review text: \textit{F. M. Markel} [J. Algebra 26, 69-74 (1973; Zbl 0274.20030)] conjectured that if \(G\) is a finite nonabelian solvable group in which no two distinct conjugacy classes have the same length, then \(G\) must be isomorphic to the symmetric group \(S_3\). Based on the analysis made by \textit{M. B. Ward} [Arch. Math. 54, 111-116 (1990; Zbl 0663.20021)] of these groups of order \(2^a3^b\) and using a battery of results about rational and \(\pi\)-rational groups, the author proves Markel's conjecture. A finite nonabelian group \(G\) is called a gdc-group if any two noncentral elements \(x\), \(y\) of \(G\) which have conjugacy classes of the same length are conjugate in \(G\). And \(G\) is called dc-group if there are no two distinct conjugacy classes in \(G\) of the same length. Stated in these terms, Markel's conjecture asserts that \(S_3\) is the unique nonabelian solvable dc-group. For finite (not necessarily solvable) groups \(G\), the author proves that a nonabelian group \(G\) is a gdc-group iff \(G\) is a dc-group. This result can be regarded as a dual result for the corresponding assertion about degrees of nonlinear characters obtained by \textit{Ya. Berkovich, D. Chillag} and \textit{M. Herzog} [Proc. Am. Math. Soc. 115, 955-959 (1992; Zbl 0822.20004)].
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E34
zbMATH DE Number: 703238
zbMATH Keywords: finite solvable groups
zbMATH Keywords: conjugacy classes
zbMATH Keywords: symmetric group \(S_3\)
zbMATH Keywords: \(\pi\)-rational groups
zbMATH Keywords: Markel's conjectures
zbMATH Keywords: gdc-groups
zbMATH Keywords: noncentral elements
zbMATH Keywords: dc-groups
author: Jiping Zhang
reviewed by: Marian Deaconescu
MaRDI profile type: Publication
full work available at URL: jabr.1994.1356
OpenAlex ID: W2073756396
DOI: 10.1006/JABR.1994.1356
description: scientific article; zbMATH DE number 703238