A countable universal torsion abelian group for purity

2022/08/29 by Ivo Herzog, Herzog, Ivo, Marcos Mazari‐Armida +1
Computer Science · Mathematics · #03C45 #03C48 #03C60 #13L05 (Secondary) #20K30 (Primary) #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2208.13913

openalex publication_date 2022/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that there is a countable universal abelian p-group for purity, i.e., a countable abelian p-group U such that every countable abelian p-group purely embeds in U. This is the last result needed to provide a complete solution to Problem 5.1 of [Fuc15] below ℵ_ω. We introduce ℵ₀-strongly homogeneous p-groups, show that there is a universal abelian p-group for purity which is ℵ₀-strongly homogeneous, and completely characterize the countable ℵ₀-strongly homogeneous p-groups.

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