When Tor(<i>A, B</i>) is a direct sum of cyclic groups

Necessary conditions are given in order that Tor(Λ, B) be a direct sum of cyclic groups for abelian groups A and B. Sufficient conditions are also given that compare with the necessary conditions.They are only slightly stronger if at all, and the two are equivalent for groups of cardinality not exceeding K 2 * Due to the complexity of the problem, these conditions are not absolute, but constitute a reduction to smaller cardinality.The results generalize earlier results of R. Nunke.What is known of the structure and properties of Tor(^4, B) for abelian groups A and B is primarily due to R. Nunke.The main results appear in [4], [5], and [6].Contributions of other authors, however, were manifested in [2] and [3].This note generalizes the results of Nunke published in [4] and [6] concerning a basic question: When is Tor(^4, B) a direct sum of cyclic groups?Nunke has provided a completely satisfactory answer when neither the cardinality of A nor B exceeds 8j and in certain other cases, as well, allowing A and B to be arbitrarily large.However, the general case where A or B has cardinality greater than # λ was deferred.In this paper, we settle the question for cardinality X 2 .For the general case, we give necessary conditions in order that Ύoτ{A, B) be a direct sum of cyclic groups, and \γe also give sufficient conditions that are very close to the necessary ones, but the gap is not bridged completely.Since Tor(^4, B) is not affected by the nontorsion portions of A and 5, we can assume without loss of generality that A and B are torsion.Further, we can specialize, as usual, to the case that A and B are both /7-primary.Therefore, all groups are assumed to be/^-primary.The following three theorems summarize the major known results concerning the question of when Tor(^4, B) is a direct sum of cyclic groups.Following [6], we say that G is Σ-cyclic if G is a direct sum of cyclic groups.THEOREM A {Nunke [4, Corollary 3.5]).If p ω A φ 0, then Tor(^, B) is Σ-cyclic only if B is Έ-cyclic.The preceding result also appears in [6] as Theorem 12(i).