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In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing thegroups which are the groups of units of commutative rings. In the followingyears, some partial answers have been given to this question in particularcases. In a previous paper \cite{DDcharp} we dealt with finite characteristicrings. In this paper we consider Fuchs' question for finite groups and weaddress this problem in two cases. Firstly, we study the case of torson-freerings and we obtain a complete classification of the finite groups of unitswhich arise in this case. Secondly, we examine the case of characteristic zerorings obtaining, a pretty good description of the possible groups of unitsequipped with families of examples of both realizable and non-realizablegroups. The main tools to deal with this general case are the Pearson andSchneider splitting of a ring \cite{PearsonSchneider70}, our previous resultson finite characteristic rings \cite{DDcharp} and our classification of thegroups of units of torsion-free rings.  As a consequence of our results we completely answer Ditor's question\cite{ditor} on the possible cardinalities of the group of units of a ring.

On Fuchs' Problem about the group of units of a ring