The classifications of countably based profinite abelian groups

In the first half of this paper, we outline the construction of a new class of abelian pro-$p$ groups, which covers all countably-based pro-$p$ groups. In the second half, we study them, and classify them up to topological isomorphism and abstract isomorphism. We use Ulm’s classification of discrete countable p-groups, which are the Pontryagin duals of such pro-$p$ groups. It emerges that they are all abstractly isomorphic to Cartesian products of finite groups and $p$-adic integers. We have thus constructed uncountably many pairwise topologically non-isomorphic profinite groups abstractly isomorphic to a Cartesian product of cyclic groups.