On fully decomposable abelian torsion groups

Abstract

A group is said to be fully decomposable, if it can be represented as the direct sum of directly indecomposable groups, the latter being groups which do not decompose into the direct sum of two of its proper subgroups. As we know that among the abelian torsion groups only the groups C(p") (0 ~ rn <--~) are directly indecomposable ([3], [6]~), the fully decomposable abelian torsion groups are those which are direct sums of cyclic and quasicyclic groups. ~ A criterion for abelian torsion groups to be fully decomposable was kown so far only for the enumerable case, in the form of a theorem of PRBFER [5] (see Corollary 6 in w 4 of the present paper). In what follows I shall give a criterion for abelian torsion groups of arbitrary power to be fully decomposable (w 2). This criterion is the generalization of a former result of mine, stating when an abelian p-group of arbitrary power is the direct sum of cyclic groups [2]. Generalizing t h i s result from a different point of view, L. FUCHS obtained recently valuable new results on groups decomposable into the direct sum of cyclic groups [1]. L. FUCHS generalized the criterion of [2], by giving a condition based, instead of the height of g roup elements, on their "relative order", the latter being closely related to the order of group elements. On the other hand, both the criterion of [2] and that of the present paper are based on the concept of the height of group elements. The criterion in [2] reads as follows: An arbitrary abelian p-grou p is the direct sum of cyclic groups if and only if the group contains no element of infinite height and there exists a principal system of elements in the group. In the present paper I show that if the concept of elements of infinite height is split in a suitable manner into that of elements of outer resp. inner infinite height, then with unchanged definition of principal system (see w 2) the same criterion holds for an arbitrary abelian p-group to be the direct