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Similar to this question, I wonder if there is some classification of all the groups which are the unit groups of some integral domain.

Since the question about rings is hard, I assume this is hard too, however perhaps the limitation to integral domains makes the question easier somehow (but if it does I don't see it).

volcanrb
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    It's a well known result that finite supgroups of unit groups of domains are cyclic. Furthermore, finite domains are fields, so I think the only finite groups which appear as unit groups of domains are cyclic groups of order $p^n -1$ with $p$ prime. – Sverre Dec 09 '20 at 10:35
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    There might be infinite domains with finite unit groups which aren't of order $p^n -1$ though. – Sverre Dec 09 '20 at 10:45

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