On a theorem of Pontrjagin

Abstract

The following criterion for countable torsion free abelian groups to be the direct sum of cyclic groups is due to PONTt~JAGIN. A countable torsion free abelian group is the direct sum of cyclic groups i f and only i f every increasing sequence of subgroups of an arbitrary finite rank r contains only a finite number of different subgroups, i Another proof of this important theorem has been given by KULIKOFF who derived it from a general result of his. ~In the present note I intend to give a short and simple proof of this theorem in a new, but plainly equivalent formulation. Let G be an additive abelian group. The letters a, b, c denote elements of G, while the other Latin small letters serve to denote rational integers. The group G is called torsion free if it contains no element ~: 0 of finite order. The elements a~ , . . . , al~ of such a group G are lineatTy independent, if an equation ml al + . . . -+m/, aT., = 0 implies ml . . . . . ml,: = 0. The maximal number of linearly independent elements of G is the rank of G. The only fact, of which I shall make use in what follows, is the basis theorem of finitely generated abelian groups, according [o which (for torsion free groups): a finitely generated torsion free abelian group is the direct sum of r cyclic groups where r is the rank of the group. Now we are going to prove the following THEOREM. A countable torsion free abelian group G is the direct sum of cyclic groups i f and only i f in G evely subgroup of finite rank is finitely generated. REMARK. On account of the cited basis theorem, this criterion may be reformulated in the following obvious manner: a countable torsion free abelian group is the direct sum of cyclic groups if and only if the same is true for