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primary groups; coproduct of cyclic groups; almost coproducts; third axiom of countability
A new class of $p$-primary abelian groups that are Hausdorff in the $p$-adic topology and that generalize direct sums of cyclic groups are studied. We call this new class of groups almost coproducts of cyclic groups. These groups are defined in terms of a modified axiom 3 system, and it is observed that such groups appear naturally. For example, $V(G)/G$ is almost a coproduct of finite cyclic groups whenever $G$ is a Hausdorff $p$-primary group and $V(G)$ is the group of normalized units of the modular group algebra over $Z/pZ$. Several results are obtained concerning almost coproducts of cyclic groups including conditions on an ascending chain that implies that the union of the chain is almost a coproduct of cyclic groups.
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