Fuchs' problem for dihedral groups

TL;DR

This paper classifies which dihedral groups can be realized as the group of units of a ring, providing a complete answer for rings of positive characteristic and certain cases of characteristic zero.

Contribution

It proves that only specific dihedral groups D_2, D_4, D_6, D_8, and D_12 occur as unit groups of rings in positive characteristic, and D_2 and D_4k (k odd) in characteristic zero.

Findings

D_2, D_4, D_6, D_8, D_12 are the only dihedral groups as units in positive characteristic.

D_2 and D_4k (k odd) are the only dihedral groups as units in characteristic zero.

Provides a complete classification for dihedral groups in this context.

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.