The classifications of countably based profinite abelian groups

TL;DR

This paper constructs and classifies a broad class of countably based abelian pro-p groups, revealing their structure as products of finite groups and p-adic integers, and distinguishes them topologically and algebraically.

Contribution

It introduces a new class of abelian pro-p groups covering all countably-based cases and classifies them using Ulm's theory, highlighting their isomorphism types.

Findings

All such groups are topologically non-isomorphic but abstractly isomorphic to products of cyclic groups.

They are classified up to topological and algebraic isomorphism using Ulm's classification.

Uncountably many non-isomorphic topological types are constructed within this class.

Abstract

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.