On Fuchs' problem for finitely generated abelian groups: The small torsion case

TL;DR

This paper classifies finitely generated abelian groups that can be realized as groups of units in rings, focusing on those with small torsion subgroups, and provides explicit classifications for certain group forms.

Contribution

It solves Fuchs' problem for finitely generated abelian groups with small torsion, including classifications of groups of the form Z/nZ × Z^r and conditions for groups of order a power of 2.

Findings

Classified groups of the form Z/nZ × Z^r for each n ≥ 2.

Established conditions for groups of order a power of 2 to be realizable.

Analyzed the structure of radical rings and their adjoint groups.

Abstract

A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such group with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Pr\"ufer rank. As a concrete instance, we classify for each $n\ge2$ the realisable groups of the form $\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}^r$. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of $2$, proving that the groups of the form $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2^{u}\mathbb{Z}$ are realisable if and only if $0\le u\le 3$ or $2^u+1$ is a Fermat's prime.