The Unit Group of a Unit Ring (the mathematical object, not your ordinary household or circus variety) is the set of all elements of a ring which are invertible (ie those elements which may be multiplied by some other element to get 1). For example note the element 1 is in the unit group of every unit ring because 1 is itself invertible. It is possible that 1 is the only invertible element in a ring (ie in the ring of integers mod 2).
The Unit Group is called such because the set of elements that make up the unit group of a ring form a group.
Not all groups can occur as the unit group of a ring, the smallest example of a group which is never the unit group of a ring is the cyclic group of order 5 (proving this amounts to a nice exercise in algebra).
Determining which groups occur as the unit groups of rings is a nontrivial problem which has been worked on by Thomas Occhipinti.
A current conjecture is that the only prime order groups which occur as the unit groups of rings are those whose order is a Mersenne prime. (Those orders clearly occur as the Unit Groups of Finite Fields).
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