Fuchs' problem for dihedral groups

Abstract

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a ring. Though progress has been made, the question remains open. One could equally well pose the question for various classes of nonabelian groups. In this paper, we prove that D_2, D_4, D_6, D_8, and D_12 are the only dihedral groups that appear as the group of units of a ring of positive characteristic (or, equivalently, of a finite ring), and D_2 and D_4k, where k is odd, are the only dihedral groups that appear as the group of units of a ring of characteristic 0.