A note on products of infinite cyclic groups

In his book [2], Fuchs introduces the notion of a subgroup X of a Specker group P being a product and goes on to establish a Lemma [2, Lemma 95.1] which yields a useful characterization of the quotient and enables an easy derivation of Nunke’s characterization of epimorphic images of the Specker group [4]. Unfortunately this Lemma is incorrect as we show in section 1. In section 2 by suitably strengthening the hypothesis we regain a characterization of the quotient. Throughout, all groups are additively written Abelian groups and our notation follows the standard works of Fuchs [1], [2].