Finite groups whose noncentral class sizes have the same \(p\)-part for some prime \(p\).
type: Item
instance of: scholarly article
title: Finite groups whose noncentral class sizes have the same \(p\)-part for some prime \(p\).
zbMATH Open document ID: 1270.20032
author: Carlo Casolo
author: Silvio Dolfi
author: Enrico Jabara
published in: Israel Journal of Mathematics
publication date: 2013-01-21T00:00:00Z
review text: A well-known theorem of Thompson states that if the prime \(p\) divides the degree of each nonlinear irreducible complex character of the finite group \(G\), then \(G\) is \(p\)-nilpotent. While it is known that this result in general has no analogue for conjugacy class sizes in place of character degrees, the paper under review presents a partial analogue as follows. If the sizes of the noncentral conjugacy classes of \(G\) have the same non-trivial \(p\)-part, then \(G\) is \(p\)-nilpotent. In addition, the authors prove that such groups are solvable of Fitting height no more than 3 and provide a lot more structural information on these groups. Moreover, they prove that if \(m\) is a positive integer such that the noncentral conjugacy class sizes \(|G:C_G(g)|\) (where \(g\in G-Z(G)\)) are all of the form \(mn_g\) for some integer \(n_g\) satisfying \((m,n_g)=1\), then \(m\) necessarily is a prime power. The authors remark that their proofs do not rely on CFSG.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6127522
zbMATH Keywords: finite groups
zbMATH Keywords: conjugacy class sizes
zbMATH Keywords: normal \(p\)-complements
zbMATH Keywords: solvability
zbMATH Keywords: Fitting heights
MaRDI profile type: Publication
full work available at URL: s11856-012-0022-3
OpenAlex ID: W2057129539
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DOI: 10.1007/S11856-012-0022-3
description: scientific article; zbMATH DE number 6127522