Fuchs’ problem for linear groups

Abstract

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.

KEYWORDS:

• Affine general linear group
• group of units
• holomorph
• realizable
• special linear group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16U60
• 20G40
• Secondary: 16S34

Additional information

Funding

The second author was supported by the Julia Gatter Hirsch Endowed Fund for Faculty/Student Research in the Mathematics Department at Gettysburg College.