Powers of irreducible characters and conjugacy classes in finite groups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1321.20030
DOI: 10.1142/S0219498814500674
publication date: 2014-09-05T00:00:00Z
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6340458
zbMATH Keywords: finite groups
zbMATH Keywords: numbers of conjugacy classes
zbMATH Keywords: irreducible characters
zbMATH Keywords: derived covering numbers
zbMATH Keywords: derived character covering numbers
author: M. R. Darafsheh
author: S. M. Robati
MaRDI profile type: Publication
cites work: Powers of characters of finite groups
cites work: Products of conjugacy classes in groups
cites work: Products of conjugacy classes and fixed point spaces
cites work: Products of conjugacy classes in finite and algebraic simple groups.
cites work: Q4103728
cites work: On finite \(p\)-groups which have only two conjugacy lengths
cites work: The covering numbers of the sporadic simple groups
cites work: Irreducible products of characters in \(A_ n\)
published in: Journal of Algebra and its Applications
title: Powers of irreducible characters and conjugacy classes in finite groups.
review text: Let \(G\) be a finite group and \(G'\) its commutator subgroup. The derived covering number \(\text{dcn}(G)\) is defined as the smallest positive integer \(n\) such that \(C^n=G'\) for all non-central conjugacy classes \(C\) of \(G\). If such an integer exists, then \(\text{dcn}(G)\) is said to be finite, otherwise infinite.NEWLINENEWLINE The authors are interested in conditions guaranteeing that \(\text{dcn}(G)\) is finite and in obtaining upper bounds in this case, and they prove some results in these directions. For example, they show that if \(G\) is nonabelian and \(G'\) is a minimal normal subgroup, then \(\text{dcn}(G)\) is finite. Also, if \(G\) is nonabelian and \(\text{dcn}(G)\) is finite, then one of the following holds: (i) \(G'\) is a minimal normal subgroup in \(G\), (ii) \(G\) is nilpotent of class 2, or (iii) \(G'\) is perfect and \(G'/Z(G')\) is minimal normal in \(G/Z(G')\), and \(Z(G')=G'\cap Z(G)\). Moreover, if \(G\) is a real finite group and \(G'\) is a nonabelian simple subgroup, then \(\text{dcn}(G)\leq 2(k(G)-1)\), where \(k(G)\) is the number of conjugacy classes of \(G\). -- Similarly the \textit{derived character covering number} is introduced to study the analogous questions for powers of characters.
reviewed by: Thomas Michael Keller
description: scientific article; zbMATH DE number 6340458