Let the functor $F\colon\bf Ring\rightarrow\bf Grp$ send the ring $A$ to its group of units $A^\times,$ and the ring homomorphism $f\colon A\rightarrow B$ to the group homomorphism $f^\times\colon A^\times\rightarrow B^\times:a\mapsto f(a)$.
I was curious about this functor, and in particular, whether it is essentially surjective. That is, for any group $G$ (not just finite,) is there a ring $A$ such that $G\cong A^\times$? If not, what groups $G$ satisfies this? A similar question was asked in this question, but what can be said for infinite groups, or groups in general? Calling groups that satisfy this condition R-groups, I have proved that any finitely generated abelian group is an R-group.
I have no idea what to do from now, but I have a conjecture that the unit group of the group ring $\mathbb F_2[G]$ is isomorphic to $G.$ If this is true, certainly, all groups are R-groups, and the functor $F$ is essentially surjective, but I am having trouble proving it. Can anyone help me?
Best Answer
No, this is already false for finite abelian groups.
A ring either has characteristic $2$ or it has a non-identity unit $-1$ which is central of order $2$, so if a group $G$ doesn't have such an element then it can only arise as the group of units of a ring of characteristic $2$.
Let $R$ be such a ring and consider an element $r \in R^{\times}$ of odd prime order $p$. (Edit: There was a sloppy argument here with an error which has now been corrected, twice!) It generates a subring of $R$ given by some quotient of the group algebra $\mathbb{F}_2[C_p]$ in which $C_p$ embeds. By Maschke's theorem $\mathbb{F}_2[C_p]$ is semisimple and hence a finite product of finite fields $\mathbb{F}_{2^k}$, and $C_p$ embeds into some $\mathbb{F}_{2^k}$ iff $p | 2^k - 1$.
So $R^{\times}$ has an element of order $2^k - 1$ where $k$ satisfies $p | 2^k - 1$. Hence:
The smallest such group is the cyclic group $C_5$ (mentioned by diracdeltafunk in the comments), which has odd order and hence no elements of order $2$, and which has an element of order $5$, but does not have an element of order $2^4 - 1 = 15$ or larger. (And the cyclic groups $C_2, C_3, C_4$ are the groups of units of the finite fields $\mathbb{F}_3, \mathbb{F}_4, \mathbb{F}_5$.)
See also the classification described by Jack Schmidt in the answer linked to by lhf in the comments.