The classifications of countably based profinite abelian groups

2011/01/15 by Jonathan Kiehlmann, Kiehlmann, Jonathan
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1101.3005

openalex publication_date 2011/01/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Effective categoricity of Abelian p-groups
• On some relatively free pro-p groups
• The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
• Procountable groups are not classifiable by countable structures
• Algebraic properties of profinite groups