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On finite-by-nilpotent profinite groups | ||
| International Journal of Group Theory | ||
| مقاله 1، دوره 9، شماره 4، اسفند 2020، صفحه 223-229 اصل مقاله (194.47 K) | ||
| نوع مقاله: Proceedings of the conference "Engel conditions in groups" - Bath - UK - 2019 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2019.119581.1577 | ||
| نویسندگان | ||
| Eloisa Detomi1؛ Marta Morigi* 2 | ||
| 1Dipartimento di Ingegneria dell'Informazione - DEI, Università di Padova, | ||
| 2Dipartimento di Matematica, Università di Bologna, Italy. | ||
| چکیده | ||
| Let $\gamma_n=[x_1,\ldots,x_n]$ be the $n$th lower central word. Suppose that $G$ is a profinite group where the conjugacy classes $x^{\gamma_n(G)}$ contains less than $2^{\aleph_0}$ elements for any $x \in G$. We prove that then $\gamma_{n+1}(G)$ has finite order. This generalizes the much celebrated theorem of B. H. Neumann that says that the commutator subgroup of a BFC-group is finite. Moreover, it implies that a profinite group $G$ is finite-by-nilpotent if and only if there is a positive integer $n$ such that $x^{\gamma_n(G)}$ contains less than $2^{\aleph_0}$ elements, for any $x\in G$. | ||
| کلیدواژهها | ||
| Conjucagy classes؛ verbal subgroups؛ profinite groups؛ FC-groups | ||
| مراجع | ||
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[1] E. Detomi, G. Donadze, M. Morig and P. Shumyatsky, On finite-by-nilpotent groups, preprint, arXiv:1907.02798.
[2] E. Detomi, B. Klopsch and P. Shumyatsky, Strong conciseness in profinite groups, preprint, arXiv:1907.01344.
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[15] A. Shalev, Profinite groups with restricted centralizers, Proc. Amer. Math. Soc., 122 (1994) 1279–1284.
[16] J. Wiegold, Groups with boundedly finite classes of conjugate elements, Proc. Roy. Soc. London Ser. A, 238 (1957) 389–401. | ||
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