Profinite groups with complemented closed subgroups

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 $G=B⋉A$ of two closed subgroups $A={\text{Cr}}_{i\in I}⟨a_i⟩$ and $B={\text{Cr}}_{j\in J}⟨b_j⟩$ that are cartesian products of cyclic groups of prime order, and with every $⟨a_i⟩$ normal in $G$. Finally, we show that a profinite-$C$ group is a $C$-group if and only if it is torsion and $|G:Z(G)\overline{G^{\prime }}|<\infty$.

10 pages