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 $A={\mathrm{Cr}}_{i\in I}〈a_i〉$ and $B={\mathrm{Cr}}_{j\in J}〈b_j〉$ 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 $|G:Z(G\left)\overline{G^{\prime }}\right|<\infty$.