The complement of a finitely generated direct summand of an abelian group

1. In his recent monograph on abelian groups [l], Kaplansky raises the following question: If F is a finitely generated abelian group and G, H are any abelian groups such that F®G=F®H, are G and H isomorphic? We shall answer this question affirmatively. In the first place we can reduce the problem to the case where F is cyclic of infinite or prime power order. For suppose that the answer has been obtained in this case, and let F be any finitely generated abelian group. Then F is a direct sum of a finite number of cyclic groups, each of infinite or prime power order, say