A Class of Primary Abelian Groups Characterized by Its Socles

Abstract

The $t$-product of a family ${\left \{ {{G_i}} \right \}_{i \in I}}$ of abelian $p$-groups is the torsion subgroup of $\prod \nolimits _{i \in I} {{G_i}}$, which we denote by $\prod \nolimits _{i \in I}^t {{G_i}}$. The $t$-product is, in the homological sense, the direct product in the category of abelian $p$-groups. Let ${\mathcal {R}^s}$ be the smallest class containing the cyclic groups that is closed with respect to direct sums, summands, and $t$-products. It is proven that two groups in ${\mathcal {R}^s}$ are isomorphic iff their socles are isomorphic as valuated vector spaces. This generalizes a classical result on direct sums of torsion-complete groups. As is frequently the case with homomorphisms defined on products, the index sets will be assumed to be nonmeasurable.