Groups with some families of complemented subgroups

A subgroup H of a group G is said to be complemented in G if there exists a subgroup K of G such that G= HK and H∩ K= 1. We prove that, for a locally soluble group G, all cyclic subgroups are complemented if and only if it is the semidirect product of groups A=Dri∈IAi by B=Drj∈JBj, where all factors Ai and Bj are finite of prime order, and A has a set of maximal subgroups normal in G with trivial intersection. An analysis of the structure of periodic locally soluble groups of infinite rank shows, in particular, that if G is a periodic locally soluble group whose infinite rank subgroups are complemented, then every subgroup of G is complemented.