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On finite groups with bounded conjugacy classes of commutators

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Representation theory of groups

Published online by Cambridge University Press:  11 August 2025

Débora Senise Open the ORCID record for Débora Senise [Opens in a new window]  and
Pavel Shumyatsky Open the ORCID record for Pavel Shumyatsky [Opens in a new window]
Débora Senise
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia, DF, Brazil
Pavel Shumyatsky*
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia, DF, Brazil
*
Corresponding author: Pavel Shumyatsky; Email: pavel2040@gmail.com
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Abstract

In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.

MSC classification

Primary: 20D25: Special subgroups (Frattini, Fitting, etc.) 20E45: Conjugacy classes 20F12: Commutator calculus

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Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 6
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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