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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A class of primary abelian groups characterized by its socles


Author: Patrick Keef
Journal: Proc. Amer. Math. Soc. 115 (1992), 647-653
MSC: Primary 20K10; Secondary 20K25
DOI: https://doi.org/10.1090/S0002-9939-1992-1101986-8
MathSciNet review: 1101986
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1101986-8
Keywords: Abelian $ p$-group, direct product, direct sum
Article copyright: © Copyright 1992 American Mathematical Society