On Neumann’s BFC-theorem and finite-by-nilpotent profinite groups

Document Type : Research Paper


Department of Mathematics, University of Brasilia, Brasilia-DF, Brazil



Let $\gamma_{n}=[x_{1},\ldots,x_{n}]$ be the $n$th lower central word and $X_{n}(G)$ the set of $\gamma_{n}$-values in a group $G$. Suppose that $G$ is a profinite group where, for each $g\in G$, there exists a positive integer $n=n(g)$ such that the set $g^{X_{n}(G)}=\{g^{y}\,|\,y\in X_{n}(G)\}$ contains less than $2^{\aleph_{0}}$ elements. We prove that $G$ is a finite-by-nilpotent group.


Main Subjects

[1] E. Detomi, G. Donadze, M. Morigi and P. Shumyatsky, On finite-by-nilpotent groups, Glasgow Mathematical Jour-
nal, 63 (2021) 54–58.
[2] E. Detomi, B. Klopsch, P. Shumyatsky, Strong conciseness in profinite groups, Journal of the London Math. Society,
102 (2020), 977-993.
[3] E. Detomi and M. Morigi, On finite-by-nilpotent profinite groups, Int. J. Group Theory, 9 no. 4 (2020) 223–229.
[4] E. Detomi, M. Morigi and P. Shumyatsky, BFC-theorems for higher commutator subgroups, Q. J. Math., 70 (2019)
[5] E. Detomi, M. Morigi and P. Shumyatsky, On conciseness of words in profinite groups, J. Pure Appl. Algebra, 220
(2016) 3010–3015.
[6] E. Detomi, M. Morigi and P. Shumyatsky, On profinite groups with commutators covered by countably many cosets,
J. Algebra, 508 (2018) 431–444.
[7] E. Detomi, M. Morigi and P. Shumyatsky, On profinite groups with word values covered by nilpotent subgroups,
Isr. J. Math., 226 (2018) 993–1008.
[8] E. Detomi, M. Morigi and P. Shumyatsky, Profinite groups with restricted centralizers of commutators, Proc. R.
Soc. Edinb., Sect. A, Math., 150 (2020) 2301–2321.
[9] G. Dierings and P. Shumyatsky, Groups with boundedly finite conjugacy classes of commutators, Q. J. Math., 69
(2018) 1047–1051.
[10] S. Franciosi, F. De Giovani and P. Shumyatsky, On groups with finite verbal conjugacy classes, Houston J. Math.,
28 (2002) 683–689.
[11] R. M. Guralnick and A. Maroti, Average dimension of fixed point spaces with applications, Adv. Math., 226 (2011)
[12] B. H. Neumann, Groups covered by permutable subsets, J. London Math. Soc., 29 (1954) 236–248.
[13] P. M. Neumann and M. R. Vaughan-Lee, An essay on BFC groups, Proc. Lond. Math. Soc., III. Ser., 35 (1977)
[14] L. Ribes and P. Zalesskii, Profinite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Berlin: Springer,
[15] D. Segal and A. Shalev, On groups with bounded conjugacy classes, Q. J. Math., Oxf. II. Ser., 50 (1999) 505–516.
[16] A. Shalev, Profinite groups with restricted centralizers, Proc. Amer. Math. Soc.,122 (1994) 1279–1284.
[17] P. Shumyatsky and W. da Silva, Engel conditions for soluble and pronilpotent groups, Advances in Group Theory
and Applications, accepted for publication, (2022).
[18] J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc. R. Soc. Lond., Ser. A, 238 (1957)