Groups whose vanishing class sizes are not divisible by a given prime.
type: Item
instance of: scholarly article
title: Groups whose vanishing class sizes are not divisible by a given prime.
zbMATH Open document ID: 1202.20010
author: Silvio Dolfi
author: Emanuele Pacifici
author: Lucia Sanus
published in: Archiv der Mathematik
publication date: 2010-05-17T00:00:00Z
review text: The authors establish that a finite group \(G\) whose vanishing conjugacy classes are not divisible by the prime \(p\) has a normal \(p\)-complement and Abelian Sylow \(p\)-subgroups. An element \(g\in G\) is a vanishing element of \(G\) if there is an irreducible complex character \(\chi\) with \(\chi(g)=0\). In this case, we define the vanishing conjugacy class of \(g\) in \(G\) to be the conjugacy class containing \(g\). The authors establish their main result in Theorem A. Note that this is an extension of Corollary A in a paper by \textit{S. Dolfi, E. Pacifici, L. Sanus} and \textit{P. Spiga} [J. Algebra 321, No. 1, 345-352 (2009; Zbl 1162.20005)] that showed if \(p\) is a prime number and the order of every vanishing element of \(G\) is not divisible by \(p\), then \(G\) has a normal Sylow \(p\)-subgroup. The proof of Theorem A relies upon the classification of the finite simple groups. The authors conclude the paper with examples. One such example shows that, for a finite group \(G\) whose vanishing conjugacy classes are not divisible by the prime \(p\), \(G/Z(G)\) may not necessarily be a \(p'\)-group or a Frobenius group.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 5708652
zbMATH Keywords: finite groups
zbMATH Keywords: vanishing elements
zbMATH Keywords: vanishing conjugacy classes
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: normal complements
zbMATH Keywords: Abelian Sylow subgroups
reviewed by: T. P. Wakefield
MaRDI profile type: Publication
full work available at URL: s00013-010-0107-3
OpenAlex ID: W2001064457
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DOI: 10.1007/S00013-010-0107-3
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description: scientific article; zbMATH DE number 5708652