TITLE: Which finite groups are the group of units of some ring?
QUESTION [44 upvotes]: 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.

REPLY [4 votes]: See also Robert W. Gilmer jr., Finite Rings having a Cyclic Group of Units, American Journal of Mathematics, Vol. 85, No. 3, July 1963.