The Schur property and groups with uniform conjugacy classes
type: Item
instance of: scholarly article
title: The Schur property and groups with uniform conjugacy classes
zbMATH Open document ID: 0831.20043
author: F. de Giovanni
author: Silvana Franciosi
author: L. A. Kurdachenko
published in: Journal of Algebra
publication date: 1995-12-11T00:00:00Z
review text: A well known theorem of Schur states that if \(G/Z(G)\) is finite then so is \(G'\). The authors say that a class of groups \(\mathfrak X\) is a Schur class if whenever \(G/Z(G) \in {\mathfrak X}\) it follows that \(G' \in {\mathfrak X}\). It is also known that polycyclic groups, Chernikov groups and soluble minimax groups are Schur classes. The first part of this paper considers classes of groups with rank restrictions showing in particular that soluble-by-finite groups of finite rank and soluble-by-finite groups with finite section rank are both Schur classes. The second part of the paper is concerned with generalizing B. H. Neumann's result that a group with boundedly finite conjugacy classes has \(G'\) finite. A group is said to have uniform \(\mathfrak X\)-conjugacy classes if there is an \(\mathfrak X\)-group \(Q\) such that \(G/C_G(x^G)\) is isomorphic to a section of \(Q\) for each \(x \in G\). If a class \(\mathfrak X\) is such that any group with uniform \(\mathfrak X\)-conjugacy classes has \(G' \in {\mathfrak X}\), then \(\mathfrak X\) must be a Schur class. However not all Schur classes satisfy this property, there being counterexamples for \(\mathfrak X\) either polycyclic or minimax. Here the authors prove the nontrivial result that the class of groups with finite section rank does have this property and, as a corollary, so does the class of Chernikov groups.
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20F19
Mathematics Subject Classification ID: 20E15
zbMATH DE Number: 783892
zbMATH Keywords: bounded conjugacy classes
zbMATH Keywords: uniform conjugacy classes
zbMATH Keywords: polycyclic groups
zbMATH Keywords: Chernikov groups
zbMATH Keywords: soluble minimax groups
zbMATH Keywords: Schur classes
zbMATH Keywords: boundedly finite conjugacy classes
zbMATH Keywords: soluble-by-finite groups of finite rank
zbMATH Keywords: groups with finite section rank
reviewed by: M. J. Tomkinson
MaRDI profile type: Publication
full work available at URL: jabr.1995.1155
OpenAlex ID: W2050881045
DOI: 10.1006/JABR.1995.1155
description: scientific article; zbMATH DE number 783892