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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2507.20791 |
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| _version_ | 1866911079748075520 |
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| author | Fernández-Alcober, Gustavo A. Sabatino, Giulia |
| author_facet | Fernández-Alcober, Gustavo A. Sabatino, Giulia |
| contents | A group $G$ is said to be a $C$-group if every subgroup $H$ has a permutable complement, i.e. if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H \cap K=1$. In this paper, we study the profinite counterpart of this concept. We say that a profinite group $G$ is profinite-$C$ if every closed subgroup admits a closed permutable complement. We first give some equivalent variants of this condition and then we determine the structure of profinite-$C$ groups: they are the semidirect products $G=B\ltimes A$ of two closed subgroups $A=\text{Cr}_{i\in I} \, \langle a_i \rangle$ and $B=\text{Cr}_{j\in J} \, \langle b_j \rangle$ that are cartesian products of cyclic groups of prime order, and with every $\langle a_i \rangle$ normal in $G$. Finally, we show that a profinite-$C$ group is a $C$-group if and only if it is torsion and $|G:Z(G)\overline{G'}|<\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_20791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Profinite groups with complemented closed subgroups Fernández-Alcober, Gustavo A. Sabatino, Giulia Group Theory 20E18, 20E15 A group $G$ is said to be a $C$-group if every subgroup $H$ has a permutable complement, i.e. if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H \cap K=1$. In this paper, we study the profinite counterpart of this concept. We say that a profinite group $G$ is profinite-$C$ if every closed subgroup admits a closed permutable complement. We first give some equivalent variants of this condition and then we determine the structure of profinite-$C$ groups: they are the semidirect products $G=B\ltimes A$ of two closed subgroups $A=\text{Cr}_{i\in I} \, \langle a_i \rangle$ and $B=\text{Cr}_{j\in J} \, \langle b_j \rangle$ that are cartesian products of cyclic groups of prime order, and with every $\langle a_i \rangle$ normal in $G$. Finally, we show that a profinite-$C$ group is a $C$-group if and only if it is torsion and $|G:Z(G)\overline{G'}|<\infty$. |
| title | Profinite groups with complemented closed subgroups |
| topic | Group Theory 20E18, 20E15 |
| url | https://arxiv.org/abs/2507.20791 |