Quasi-isomorphism of direct sums of cyclic groups

Abstract

F o r any ordinal number e, fG(e) is the e-th Ulm invariant of G. Conversely, it was shown that if G and H are countable groups which satisfy {I) and (II), then G H. Clearly, for countable groups without elements of infinite height, (I) is a necessary and sufficient condition for quasi-isomorphism. As a corollary of [1; Theorem 1.3] we have that if G and H are countable, then G H if and only if G and H have quasi-isomorphic basic subgroups and G t H t (G ~ and H t are the subgroups of dements of infinite height in G and H respectively). In this note, we use [1; Theorem 1.3] to prove the following theorem.