Profinite groups with complemented closed subgroups

TL;DR

This paper characterizes profinite groups where every closed subgroup has a permutable closed complement, revealing their structure as semidirect products of cyclic groups of prime order and linking them to classical $C$-groups.

Contribution

It provides a structural classification of profinite-$C$ groups, extending the concept of $C$-groups to the profinite setting and establishing conditions for when they are classical $C$-groups.

Findings

Profinite-$C$ groups are semidirect products of cyclic prime order groups.

They are characterized by being torsion with finite index over their center and derived subgroup.

Equivalent variants of the profinite-$C$ condition are established.

Abstract

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 $|G:Z(G)\overline{G'}|<\infty$.