### Invariants on primary abelian groups and a problem of Nunke

#### Abstract

If is an arbitrary abelian -group, an invariant is defined which measures how closely resembles a direct sum of cyclic groups. This invariant consists of a class of finite sets of regular cardinals, and is inductively constructed using filtrations of various subgroups of ; can also be considered to be a measure of the presence of non-zero elements of infinite height in . This construction is particularly useful when the group has final rank less than the smallest weakly Mahlo cardinal; and in this case, is a direct sum of cyclics iff is empty. These deliberations are then used to place several of the most significant results relating to direct sums of cyclics into a significantly broader context. For example, is shown to be almost a direct sum of cyclics iff every set in has at least two elements. Finally, is used to give a more complete and concrete answer to a classical problem of Nunke, which asks when the torsion product of two abelian -groups is a direct sum of cyclics.

primary abelian groups; direct sums of cyclics; invariants; filtrations; torsion product; almost direct sums of cyclics

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