A converse of Baer's theorem.
type: Item
instance of: scholarly article
title: A converse of Baer's theorem.
zbMATH Open document ID: 1311.20037
author: Rasoul Hatamian
author: Mitra Hassanzadeh
author: Saeed Kayvanfar
published in: Ricerche di Matematica
publication date: 2014-12-08T00:00:00Z
review text: Schur showed for any group \(G\) that the finiteness of the quotient group \(G/Z(G)\) implies that of the derived subgroup \(G'\). Baer generalized this to the assertion that the finiteness of \(G/Z_n(G)\) implies that of \(\gamma_{n+1}(G)\). Here, \(\{Z_n\}\) and \(\{\gamma_n\}\) are the upper and lower central series, respectively. Philip Hall proved that the converse of Baer's result is false; he showed that finiteness of \(\gamma_{n+1}(G)\) implies that of \(G/Z_{2n}(G)\) but not necessarily that of \(G/Z_n(G)\). Hekster proved a converse of Baer's theorem under the assumption that \(G\) is finitely generated. In the present paper, the authors prove the stronger result that the converse to Baer's theorem holds under the weaker assumption that \(G/Z_n(G)\) is finitely generated. In fact, they prove the precise version: Theorem. Let \(G\) be a group such that \(G/Z_n(G)\) is finitely generated and \(\gamma_{n+1}(G)\) is finite. Then, \[ |G/Z_n(G)|\leq|\gamma_{n+1}(G)|^{d(G/Z_n(G))}. \] Here, \(d(H)\) denotes the minimal number of generators for a group \(H\). The main idea they employ is that of isoclinism which makes many such proofs simple. For groups \(G\) as in the theorem above which satisfy the additional assumption that \(G\) is nilpotent, the authors deduce the interesting corollary that \(|G/Z_n(G)|\) actually divides \(|\gamma_{n+1}(G)|^{d(G/Z_n(G))}\).
reviewed by: Balasubramanian Sury
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6378265
zbMATH Keywords: converse of Baer theorem
zbMATH Keywords: central series
zbMATH Keywords: isoclinism
zbMATH Keywords: commutator subgroup
zbMATH Keywords: central factors
zbMATH Keywords: derived subgroup
zbMATH Keywords: finitely generated groups
zbMATH Keywords: numbers of generators
MaRDI profile type: Publication
full work available at URL: s11587-013-0172-6
OpenAlex ID: W2131831613
cites work: Endlichkeitskriterien für Kommutatorgruppen
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DOI: 10.1007/S11587-013-0172-6
description: scientific article; zbMATH DE number 6378265