Finite groups of units of finite characteristic rings

TL;DR

This paper characterizes the groups of units in finite characteristic rings, providing a near-complete classification and answering related questions about possible unit group sizes.

Contribution

It offers a detailed description of unit groups in finite characteristic rings and classifies all possible sizes of these groups.

Findings

Complete characterization of unit groups in finite characteristic rings

Classification of all possible cardinalities of unit groups

Examples illustrating obstacles to full classification

Abstract

In \cite[Problem 72]{Fuchs60} Fuchs asked the following question: which groups can be the group of units of a commutative ring? In the following years, some partial answers have been given to this question in particular cases. The aim of the present paper is to address Fuchs' question when $A$ is a {\it finite characteristic ring}. The result is a pretty good description of the groups which can occur as group of units in this case, equipped with examples showing that there are obstacles to a "short" complete classification. As a byproduct, we are able to classify all possible cardinalities of the group of units of a finite characteristic ring, so to answer Ditor's question \cite{ditor}.