Vol. 42, No. 1, 1972

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

Paul Daniel Hill

Vol. 42 (1972), No. 1, 63–67

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 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
Received: 17 February 1970
Revised: 14 December 1971
Published: 1 July 1972
Paul Daniel Hill