The size of the commutator subgroup of finite groups.
type: Item
instance of: scholarly article
title: The size of the commutator subgroup of finite groups.
zbMATH Open document ID: 1157.20013
author: Marcel Herzog
author: Gil Kaplan
author: Arieh Lev
published in: Journal of Algebra
publication date: 2008-10-21T00:00:00Z
review text: Let \(G\) be a finite group. In the paper [J. Algebra 278, No. 2, 494-501 (2004; Zbl 1060.20018)] of the authors it was proved that if the group \(G\) is nontrivial solvable with trivial Frattini subgroup and with trivial center, then the order of the commutator subgroup is greater than the cube root of the order of \(G\). The aim of the paper is to prove that if the group \(G\) is nonabelian with trivial Frattini subgroup, then the order of the commutator subgroup is greater than the square root of the order of the central factor \(G/Z(G)\). The example of a Frobenius group of order \(p(p-1)\) with \(p\) a prime number shows that the inequality cannot be improved in general. -- The proof uses the \(k(GV)\)-theorem concerning the number of conjugacy classes of the semidirect product \(GV\) established recently by \textit{D. Gluck, K. Magaard, U. Riese, P. Schmid} [in J. Algebra 279, No. 2, 694-719 (2004; Zbl 1058.20010)].
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5354918
zbMATH Keywords: commutator subgroup
zbMATH Keywords: center
zbMATH Keywords: Frattini subgroup
zbMATH Keywords: finite groups
zbMATH Keywords: solvable groups
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.12.036
OpenAlex ID: W2013298023
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DOI: 10.1016/J.JALGEBRA.2005.12.036
description: scientific article; zbMATH DE number 5354918