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Proceedings of the American Mathematical Society

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



Countable torsion products of abelian $ p$-groups

Author: E. L. Lady
Journal: Proc. Amer. Math. Soc. 37 (1973), 10-16
MSC: Primary 20K25
MathSciNet review: 0313420
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Abstract: Let $ {A_1},{A_2}, \cdots $ be direct sums of cyclic p-groups, and $ G = t\prod\nolimits_1^\infty {{A_n}} $ be the torsion subgroup of the product. The product decomposition is used to define a topology on G in which each $ G[{p^r}]$ is complete and the restriction to $ G[{p^r}]$ of any endomorphism is continuous. Theorems are then derived dual to those proved by Irwin, Richman and Walker for countable direct sums of torsion complete groups. It is shown that every subgroup of $ G[p]$ supports a pure subgroup of G, and that if $ G = H \oplus K$, then $ H \approx t\prod\nolimits_1^\infty {{G_n}} $, where the $ {G_n}$ are direct sums of cyclics. A weak isomorphic refinement theorem is proved for decompositions of G as a torsion product. Finally, in answer to a question of Irwin and O'Neill, an example is given of a direct decomposition of G that is not induced by a decomposition of $ \prod\nolimits_1^\infty {{A_n}} $.

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Keywords: Abelian p-groups, direct sum of cyclics, socle, direct product, pure complete group
Article copyright: © Copyright 1973 American Mathematical Society