Countable torsion products of abelian $p$-groups

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 $\prod \nolimits _1^\infty {{A_n}}$.