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Journal Article

# A Class of Primary Abelian Groups Characterized by Its Socles

Patrick Keef
Proceedings of the American Mathematical Society
Vol. 115, No. 3 (Jul., 1992), pp. 647-653
DOI: 10.2307/2159210
Stable URL: http://www.jstor.org/stable/2159210
Page Count: 7

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Topics: Vector spaces, Homomorphisms, Direct products, Integers, College mathematics, Isomorphism, Algebra

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## Abstract

The t-product of a family {Gi}i ∈ I of abelian p-groups is the torsion subgroup of $\prod_{i \in I}G_i$, which we denote by $\prod^t_{i \in I} G_i$. The t-product is, in the homological sense, the direct product in the category of abelian p-groups. Let Rs 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 Rs 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.

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