Abstract
A group G is said to have restricted centralizers if for each g in G the centralizer \(C_G(g)\) either is finite or has finite index in G. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes \(\pi \), we take interest in profinite groups with restricted centralizers of \(\pi \)-elements. It is shown that such a profinite group has an open subgroup of the form \(P\times Q\), where P is an abelian pro-\(\pi \) subgroup and Q is a pro-\(\pi '\) subgroup. This significantly strengthens a result from our earlier paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acciarri, C., Shumyatsky, P.: A stronger form of Neumann’s BFC-theorem. Isr. J. Math. 242, 269–278 (2021). https://doi.org/10.1007/s11856-021-2133-1
Detomi, E., Shumyatsky, P.: On the commuting probability for subgroups of a finite group, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 1–14 (2021). https://doi.org/10.1017/prm.2021.68
Detomi, E., Morigi, M., Shumyatsky, P.: Profinite groups with restricted centralizers of commutators. Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 150(5), 2301–2321 (2020). https://doi.org/10.1017/prm.2019.17
Dierings, G., Shumyatsky, P.: Groups with boundedly finite conjugacy classes of commutators. Quarterly J. Math. 69(3), 1047–1051 (2018)
Gorenstein, D.: Finite Groups. Chelsea Publishing Company, New York (1980)
Kelley, J. L.: General topology, Grad. Texts in Math., vol. 27, Springer, New York (1975)
Khukhro, E.I.: Nilpotent groups and their automorphisms. de Gruyter, Berlin-New York (1993)
Neumann, B.H.: Groups covered by permutable subsets. J. London Math. Soc. (3) 29, 236–248 (1954)
Neumann, P.M.: Two combinatorial problems in group theory. Bull. Lond. Math. Soc. 21, 456–458 (1989)
Ribes, L., Zalesskii, P.: Profinite Groups, 2nd edn. Springer Verlag, Berlin, New York (2010)
Robinson, D. J. S.: A course in the theory of groups, Second edition. Graduate Texts in Mathematics, 80. Springer-Verlag, New York (1996)
Rowley, P.: Finite groups admitting a fixed-point-free automorphism group. J. Algebra 174, 724–727 (1995)
Shalev, A.: Profinite groups with restricted centralizers. Proc. Amer. Math. Soc. 122, 1279–1284 (1994)
Wiegold, J.: Groups with boundedly finite classes of conjugate elements. Proc. Roy. Soc. London Ser. A 238, 389–401 (1957)
Wilson, J.S.: Profinite Groups. Clarendon Press, Oxford (1998)
Zelmanov, E.I.: On periodic compact groups. Israel J. Math. 77, 83–95 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Apoio à Pesquisa do Distrito Federal (FAPDF), Brazil.
Rights and permissions
About this article
Cite this article
Acciarri, C., Shumyatsky, P. Profinite groups with restricted centralizers of \(\pi \)-elements. Math. Z. 301, 1039–1045 (2022). https://doi.org/10.1007/s00209-021-02955-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-021-02955-9