Common transversals and complements in abelian groups

Given a finite abelian group G and cyclic subgroups A , B , C of G of the same order, we find necessary and sufficient conditions for A , B , C to admit a common transversal for the cosets they afford. For an arbitrary number of cyclic subgroups, we give a sufficient criterion when there exists a common complement. Moreover, in several cases where a common transversal exists, we provide concrete constructions.