Department of Mathematics
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For some long-forgotten reason I needed a particular example of a (noetherian) ring with a certain small group of units. Along the way, I showed, somewhat by accident, that no ring has Z/5Z as its group of units.

Somewhat later, having consulted with some ring-theorists on the question of which groups could be (isomorphic to) the group of units of some ring, I was quite surprised to learn that this is a deep question with no satisfactory answers: there are no broad, useful general necessary or sufficient conditions known on a group that it be the group of units of some ring.

Since then I have approached this somewhat as a "recreational mathematics" topic. I have not done any sort of literature search. I will report to you on some examples and conjectures (for instance, I conjecture that if m is congruent to 5 mod 6, then Z/mZ is not the group of units of a ring). Many of these problems appear to be of a combinatorial or number theoretic nature, rather than ring-theoretic.

I hope to have an undergraduate research student working on this for me this summer: in particular, starting with that all-important literature search.