Groups in which squares have boundedly many conjugates
type: Item
instance of: scholarly article
title: Groups in which squares have boundedly many conjugates
zbMATH Open document ID: 1439.20028
author: Gláucia Dierings
author: Pavel Shumyatsky
published in: Journal of Group Theory
publication date: 2019-01-09T00:00:00Z
review text: The authors are interested in the implications of bounding the sizes of certain conjugacy classes in a group. An early result in this direction is due to \textit{B. H. Neumann} [J. Lond. Math. Soc. 29, 236--248 (1954; Zbl 0055.01604)] and says that if all conjugacy class sizes in a group \(G\) are finite and of bounded size then the derived group \(G'\) is finite. In particular, denoting the conjugacy class of \(x \in G\) by \(x^G\), if \(|x^G| \leq n\) for all \(x \in G\) then \(|G'|\) is \(n\)-bounded. The authors have previously considered the case when conjugacy class sizes of commutators are bounded by \(n\) and concluded that in this case \(G''\) is finite and its order is \(n\)-bounded. In the present paper, the authors continue this investigation, in particular they consider the implications of bounding the conjugacy class sizes of square elements. An element \(x \in G\) is called a square if there exists \(y \in G\) such that \(x = y^2\). The main result of the paper under review is the following. Let \(H\) denote the subgroup generated by all square elements and suppose \(|x^H| \leq n\) for any square \(x \in G\). Then \(\gamma_3(H)\), the third term of the lower central series of \(H\), is finite and \(n\)-bounded. The authors leave as an open question whether \(H'\) is finite and \(n\)-bounded if \(|x^G| \leq n\) for any square \(x \in G\).
reviewed by: Rachel D. Camina
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20F14
zbMATH DE Number: 6998942
zbMATH Keywords: conjugacy classes
zbMATH Keywords: lower central series
zbMATH Keywords: derived series
MaRDI profile type: Publication
full work available at URL: jgth-2018-0110
OpenAlex ID: W2879876301
cites work: Groups with boundedly finite conjugacy classes of commutators
cites work: Average dimension of fixed point spaces with applications.
cites work: Groups Covered By Permutable Subsets
cites work: An Essay on BFC Groups
cites work: On groups with bounded conjugacy classes
cites work: Groups with boundedly finite classes of conjugate elements
DOI: 10.1515/JGTH-2018-0110
Wikidata QID: Q129533976
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description: scientific article; zbMATH DE number 6998942