Hostname: page-component-cb9f654ff-5kfdg Total loading time: 0 Render date: 2025-08-16T13:24:32.411Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Acciarri, C. and Shumyatsky, P., A stronger form of Neumann’s BFC-theorem, Israel J. Math. 242 (2021), 269–278.10.1007/s11856-021-2133-1CrossRefGoogle Scholar
Detomi, E., Morigi, M. and Shumyatsky, P., BFC-theorems for higher commutator subgroups, Quart. J. Math. 70 (2019), 849–858.10.1093/qmath/hay068CrossRefGoogle Scholar
Dierings, G. and Shumyatsky, P., Groups with boundedly finite conjugacy classes of commutators, Quart. J. Math. 69 (2018), 1047–1051.10.1093/qmath/hay014CrossRefGoogle Scholar
Doerk, K. and Hawkes, T., Finite soluble groups (de Gruyter, Berlin, 1992).10.1515/9783110870138CrossRefGoogle Scholar
Eberhard, S. and Shumyatsky, P., Probabilistically nilpotent groups of class two, Math. Ann. 388 (2024), 1879–1902.10.1007/s00208-023-02567-0CrossRefGoogle Scholar
Feit, W. and Thompson, J. G., Solvability of groups of odd order, Pacific J. Math. 13 (1963), 773–1029.Google Scholar
Gorenstein, D., Finite groups (Chelsea, New York, 1980).Google Scholar
Guralnick, R. M. and Maroti, A., Average dimension of fixed point spaces with applications, J. Algebra 226 (2011), 298–308.Google Scholar
Khukhro, E. I., Nilpotent groups and their automorphisms (De Gruyter, New York, 1993).10.1515/9783110846218CrossRefGoogle Scholar
Neumann, B. H., Groups covered by permutable subsets, J. London Math. Soc. 29 (1954), 236–248.10.1112/jlms/s1-29.2.236CrossRefGoogle Scholar
Neumann, P. M. and Vaughan-Lee, M. R., An essay on BFC groups, Proc. Lond. Math. Soc. 35 (1977), 213–237.10.1112/plms/s3-35.2.213CrossRefGoogle Scholar
Robinson, D. J. S., A course in the theory of groups, Graduate Texts in Mathematics , vol. 80, 2nd edition (Springer-Verlag, New York, 1996).10.1007/978-1-4419-8594-1CrossRefGoogle Scholar
Segal, D. and Shalev, A., On groups with bounded conjugacy classes, Quart. J. Math. 50 (1999), 505–516.10.1093/qjmath/50.200.505CrossRefGoogle Scholar
Shumyatsky, P., Bounded conjugacy classes, commutators and approximate subgroups, Quart. J. Math. 73 (2022), 679–684.10.1093/qmath/haab049CrossRefGoogle Scholar
Wiegold, J., Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London Ser. A 238 (1957), 389–401.Google Scholar