Coprime commutators in profinite groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866908940377260032 |
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| author | Acciarri, Cristina Shumyatsky, Pavel |
| author_facet | Acciarri, Cristina Shumyatsky, Pavel |
| contents | By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent residual $γ_\infty(G)$. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of $γ_\infty(G)$ and, more generally, on the structure of $G$. In this paper we show that if the set of coprime commutators of a profinite group $G$ is covered by countably many procyclic subgroups, then $γ_\infty(G)$ is finite-by-procyclic. In particular, it follows that $G$ is finite-by-pronilpotent-by-abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03754 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Coprime commutators in profinite groups Acciarri, Cristina Shumyatsky, Pavel Group Theory 20E18 By a coprime commutator in a profinite group $G$ we mean any element of the form $[x, y]$, where $x,y\in G$ and $(|x|,|y|)=1$. It is well-known that the subgroup generated by the coprime commutators of $G$ is precisely the pronilpotent residual $γ_\infty(G)$. There are several recent works showing that finiteness conditions on the set of coprime commutators have strong impact on the properties of $γ_\infty(G)$ and, more generally, on the structure of $G$. In this paper we show that if the set of coprime commutators of a profinite group $G$ is covered by countably many procyclic subgroups, then $γ_\infty(G)$ is finite-by-procyclic. In particular, it follows that $G$ is finite-by-pronilpotent-by-abelian. |
| title | Coprime commutators in profinite groups |
| topic | Group Theory 20E18 |
| url | https://arxiv.org/abs/2510.03754 |