Which finite groups are the group of units of some ring?

Motivated by this question:

• Which finite groups are the group of units of some ring?

• Which finite groups are the group of units of some finite ring?

• Which finite abelian groups are the group of units of some commutative ring?

• Which finite abelian groups are the group of units of some finite commutative ring?

It seems that even for fields this is not simple to answer.

I expect the general question (Which groups are the group of units of some ring?, with no finiteness hypotheses) is even harder.

I searched the literature and found that the cyclic groups are completely classified. If one restricts one's attention to rings with torsion-free additive groups, then the answer is know fairly precisely (all ingredients are known, but the classification of recipes is incomplete). I include a different kind of result on division rings, as it is used in the torsion-free case and is interesting at any rate.

I suspect the literal answers to all of your questions is that we do not know.

Cyclic groups

Pearson–Schneider (1970) classify those cyclic groups that occur as groups of units of rings. They are exactly the infinite cyclic group and the finite cyclic groups whose order can be written as a coprime product of numbers of the following forms (forms may be repeated):

• $q^t-1$, $q$ prime, $t \geq 1$
• $q^s(q-1)$, $q$ and odd prime, $s \geq 1$;
• $4m+2$, $m \geq 0$
• $4n$, $n$ odd positive and if a prime $p$ divides $n$, then $1 \equiv p \mod 4$.

The paper is:

• Pearson, K. R.; Schneider, J. E. “Rings with a cyclic group of units.” J. Algebra 16 (1970) 243–251. MR265405 DOI:10.1016/0021-8693(70)90030-X

Division rings

Amitsur (1955) classified the finite subgroups of division rings. Note that the group of units of a non-commutative division ring is never itself finite, so this is not directly relevant to the question (but I suspect is interesting to readers).

• Amitsur, S. A. “Finite subgroups of division rings.” Trans. Amer. Math. Soc. 80 (1955), 361–386. MR74393 DOI:10.2307/1992994

Torsion-free rings

Amitsur's work was generalized to rings whose additive groups are torsion-free by Hirsch–Zassenhaus (1966). Here the groups must be built in a simple way from just a few small groups: C2, C4, C6, Q8, BT24, and DC12. The automorphism group of the additive group is the group of units of the endomorphism ring of the additive group, and conversely Corner has showed that every countable ring with a torsion-free reduced additive group is an endomorphism ring of a torsion-free abelian group. Hence the finite groups involved are the same.

• Corner, A. L. S. “Every countable reduced torsion-free ring is an endomorphism ring.” Proc. London Math. Soc. (s3) 13 (1963) 687–710. MR153743 DOI:10.1112/plms/s3-13.1.687

• Hirsch, K. A.; Zassenhaus, H. “Finite automorphism groups of torsion-free groups.” J. London Math. Soc. 41 (1966) 545–549. MR197549 DOI:10.1112/jlms/s1-41.1.545

See also Robert W. Gilmer jr., Finite Rings having a Cyclic Group of Units, American Journal of Mathematics, Vol. 85, No. 3, July 1963.