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 boolean AND 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 = Dr(i is an element of I) A(i) by B = Dr(j is an element of J) B-j, where all factors A(i) and B-j 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.