A character theoretic criterion for a \(p\)-closed group.
type: Item
instance of: scholarly article
title: A character theoretic criterion for a \(p\)-closed group.
zbMATH Open document ID: 1264.20011
author: Guohua Qian
published in: Israel Journal of Mathematics
publication date: 2012-11-13T00:00:00Z
review text: By a result of \textit{S. M. Gagola} jun. and \textit{M. L. Lewis} [Commun. Algebra 27, No. 3, 1053-1056 (1999; Zbl 0929.20010)] a finite group \(G\) is nilpotent if and only if \(\chi(1)^2\) divides \(|G/\ker\chi|\) for any irreducible character \(\chi\) of \(G\). Now let \(p\) be a prime, and for an integer \(n\) denote by \(n_p\) the largest power of \(p\) dividing \(n\). The author obtains the following \(p\)-local version (and thus generalization) of the Gagola-Lewis result. Suppose that \(|G/\ker\chi|_p\geq(\chi(1)^2)_p\) for all \(\chi\in\text{Irr}(G)\). Then one of the following holds: (1) \(G\) has a normal Sylow \(p\)-subgroup; (2) \(p=3\), and \(G\) has a composition factor \(A_7\); (3) \(p=2\), and \(G\) has a composition factor belonging to \(\{A_7,A_{11},A_{13},M_{22}\}\). As an immediate consequence of this result, a finite \(p\)-solvable group \(G\) has a normal Sylow \(p\)-subgroup if and only if \(|G/\ker\chi|_p\geq(\chi(1)^2)_p\) for all \(\chi\in\text{Irr}(G)\). A couple of corollaries are also proved, for example, the finite group \(G\) is 3-closed if and only if \(|G/\ker\chi|_3>(\chi(1)^2)_3\) for every \(\chi\in\text{Irr}(G)\) whose degree is divisible by 3.
reviewed by: Thomas Michael Keller
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6105517
zbMATH Keywords: finite groups
zbMATH Keywords: \(p\)-closed groups
zbMATH Keywords: character criteria
zbMATH Keywords: 3-closed groups
zbMATH Keywords: irreducible characters
zbMATH Keywords: normal Sylow subgroups
zbMATH Keywords: composition factors
MaRDI profile type: Publication
full work available at URL: s11856-011-0198-y
OpenAlex ID: W1982022828
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DOI: 10.1007/S11856-011-0198-Y
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description: scientific article; zbMATH DE number 6105517