A subnormality criterion in finite groups related to character degrees
type: Item
instance of: scholarly article
title: A subnormality criterion in finite groups related to character degrees
zbMATH Open document ID: 0915.20004
author: Udo Riese
published in: Journal of Algebra
publication date: 1999-07-07T00:00:00Z
review text: Suppose that \(A\) is a proper abelian subgroup of a finite group \(G\). Then, if \(\chi\in\text{Irr}(G)\), then \(\chi(1)\leq| G:A|\). One proves the following important assertion (Theorem A): If \(\chi(1)=| G:A|\), then \(A\) is subnormal in \(G\). The author shows that \(A\) need not be normal in \(G\). Note that even if \(A\) is normal in \(G\) and \(| G:A|\in\text{cd}(G)\), then \(G/A\) may be an arbitrary group. Theorem A cannot be extended to arbitrary nilpotent subgroups. The following theorem is an immediate consequence of Theorem A. Theorem B. Suppose that \(A\) is an abelian subgroup of \(G\). If \(| G:A|\in\text{cd}(G)\), then \(G_0=A^G\), the normal closure of \(A\) in \(G\), is a nilpotent group of central type whose center \(Z_0\) satisfies \(| A:Z_0|=| G_0:A|\). Moreover, \(C_G(Z_0)=G_0\) and \(| G:G_0|<| Z_0|\). The author poses the following Problem. Suppose the finite group \(G\) has a proper nilpotent subgroup \(H\) possessing a linear character \(\lambda\) such that \(\lambda^G\) is irreducible. Can \(G\) be a simple group? I think that this problem can be solved using the classification of finite simple groups. For example, if \(H\in\text{Syl}(G)\), then Chillag and Herzog proved that \(G\) in fact is not simple (moreover, in the last case, the Fitting subgroup of \(G>\{1\}\) is nonidentity).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1214104
zbMATH Keywords: finite groups
zbMATH Keywords: subnormal subgroups
zbMATH Keywords: Abelian subgroups
zbMATH Keywords: nilpotent subgroups
reviewed by: Yakov Berkovich
MaRDI profile type: Publication
full work available at URL: jabr.1997.7259
OpenAlex ID: W1974570940
cites work: Finite groups with an irreducible representation of large degree
cites work: On groups of central type
cites work: Q4103728
cites work: Criteria for subnormality in finite groups.
cites work: Q3907739
cites work: Q3256242
DOI: 10.1006/JABR.1997.7259
description: scientific article; zbMATH DE number 1214104