Perhaps this is a trivial question, but I have no idea how to justify it.


Call a pair of groups $(G\_1, G\_2)$ ring-compatible if $G\_1$ is abelian and there exists a ring $R$ with addition and multiplication $(+, \ast)$ say satisfying the condition that $(R, +)$ is isomorphic to $G\_1$ and the group of units of $R$, say $U(R)$, satisfy $(U(R), \ast) \cong G\_2$.


For instance, the existence of finite fields of order $p^n$ for a prime $p$ implies that the pair $(G\_1, G\_2)$ with $G\_1$ being the group $(C\_p)^n = C\_p \times \cdots \times C\_p$ and $G\_2$ being the cyclic group of order $p^n - 1$ is a ring-compatible pair. 


Are all pairs of groups $(G\_1, G\_2)$ with $G\_1$ abelian and if $G\_1, G\_2$ are finite groups, $|G\_1| > |G\_2$, ring compatible?


I apologize in advance if this question is trivial.



A basic question about rings
The answer for your question is NO, not every pair of groups is ring compatible, even if the first group is abelian. There are abelian groups $G\_1$ (called Nil groups), in which only zero multiplication turns $G\_1$ into a ring. A complete discussion of your question can be find in
"L. Fuchs, Infinite abelian groups"
(Despite the name of book, you can find facts about additive group of finite rings)


