  Which groups can occur as the group of units in a ring? Such groups are
called realizable. Though the realizable members of several classes of groups
have been determined (e.g., cyclic, odd order, alternating, symmetric, finite
simple, indecomposable abelian, and dihedral), the question remains open. The
general linear groups are realizable by definition: they are the units in the
corresponding matrix rings. In this paper, we study the realizability of two
closely related linear groups, the special linear groups and the affine general
linear groups. We determine which special linear groups of degree 2 over a
finite field are realizable by a finite ring, and we determine which affine
general linear groups of degree 1 over a cyclic group are realizable by a
finite ring. We also give partial results for certain linear groups of other
degrees and for rings of characteristic zero.
