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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.

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Related question: math.stackexchange.com/questions/73498/…. –  lhf May 7 '13 at 12:44
    
You can work out the complete answer for the last question (which obviously reduces to the cyclic case) using the related question in your comment. –  Martin Brandenburg May 7 '13 at 13:33
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@Martin: The group of units of a primary commutative ring need not be cyclic, so I think the reduction is only halfway (sufficient conditions). –  Jack Schmidt May 7 '13 at 14:25
    
@JackSchmidt Yes, I deleted my comment presumably just as you were posting yours. There is a result I was thinking of...I shall see if I can find it. –  user1729 May 7 '13 at 14:27
    
(I think that I was thinking "Every linear group can be embedded into a subgroup of a group of units of a ring". This does not hold for some groups, but allows us to generalise linear groups to this setting. I cannot remember the generalisation though...) –  user1729 May 7 '13 at 14:30
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2 Answers 2

up vote 12 down vote accepted

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:

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

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This is an excellent resource. –  Alexander Gruber May 7 '13 at 20:25
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See also Robert W. Gilmer jr., Finite Rings having a Cyclic Group of Units, American Journal of Mathematics, Vol. 85, No. 3, July 1963.

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This paper is very good, but focusses on the rings, rather than the groups. It is cited by Pearson–Schneider (1970), which reproves the ring classification results (and generalizes them though not strongly) while it classifies the groups. –  Jack Schmidt May 7 '13 at 16:26
    
Jack, thanks for this clarification! I read Gilmer's paper a long time ago and wasn't aware of Pearson-Schneider's. –  Nicky Hekster May 7 '13 at 20:34
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