The converse of Schur's theorem
type: Item
instance of: scholarly article
title: The converse of Schur's theorem
zbMATH Open document ID: 1197.20023
author: Peyman Niroomand
published in: Archiv der Mathematik
publication date: 2010-06-04T00:00:00Z
review text: Schur's theorem states that for a group \(G\) finiteness of the central factor-group implies the finiteness of the derived subgroup. The converse was proved by \textit{P. Hilton} [Int. J. Math. Math. Sci. 28, No. 8, 455-460 (2001; Zbl 1050.20021)] in case \(G/Z(G)\) is finitely generated. The author establishes that in such a case, \(|G/Z(G)|\leq |G'|^{d(G/Z(G))}\) and, provided the group \(G\) is nilpotent, \(|G/Z(G)|\) divides \(|G'|^{d(G/Z(G))}\).
reviewed by: János Kurdics
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 5716890
zbMATH Keywords: Schur theorem
zbMATH Keywords: finite groups
zbMATH Keywords: center
zbMATH Keywords: central factor
zbMATH Keywords: derived subgroup
zbMATH Keywords: nilpotent groups
zbMATH Keywords: finitely generated groups
MaRDI profile type: Publication
full work available at URL: s00013-010-0106-4
OpenAlex ID: W2071655091
cites work: On a theorem of Schur.
cites work: On variants of a theorem of Schur.
DOI: 10.1007/S00013-010-0106-4
description: scientific article; zbMATH DE number 5716890