Primary groups whose subgroups of smaller cardinality are direct sums of cyclic groups

Paul Daniel Hill

Vol. 42 (1972), No. 1, 63–67

DOI: 10.2140/pjm.1972.42.63

Abstract

Let G denote a primary abelian group. The conjecture (partially supported by a theorem of Nunke) that there exists, for each infinite cardinal m, a group G of cardinality m that is not a direct sum of cyclic groups but has the property that each subgroup of G having cardinality less than m is a direct sum of cyclic groups is shown to be false. More specifically, it is shown that if a primary group has cardinality ℵ_ω and each subgroup of smaller cardinality is a direct sum of cyclic groups, then so is the group.

Further, we show that if G is the union of a countable chain of pure subgroups, then G is a direct sum of cyclic groups if and only if the subgroups in the given chain are direct sums of cyclic groups.

Mathematical Subject Classification 2000

Primary: 20K25

Milestones

Received: 17 February 1970

Revised: 14 December 1971

Published: 1 July 1972