Fuchs' problem for linear groups
Keir Lockridge, Jacinda Terkel
TL;DR
The paper investigates which natural linear groups can occur as unit groups of unital rings, focusing on $\mathbf{SL}_2(\mathbf{F}_q)$ and $\mathbf{AGL}_1(\mathbf{Z}_n)$. It derives sharp characteristic-dependent constraints for finite rings, showing, for example, that $\mathbf{SL}_2(\mathbf{F}_q)$ is realizable only in restricted cases (e.g., $q=2$, or specific small exceptional situations), and that many $\mathbf{SL}_2(\mathbf{F}_q)$ with odd $q>3$ are not finite-ring realizable; in characteristic zero a Hurwitz-integers example realizes $\mathbf{SL}_2(\mathbf{F}_3)$ while many other cases are obstructed by maximal abelian subgroups. For $\mathbf{Hol}(\mathbf{Z}_n)$, the authors provide a complete table of characteristic-dependent realizability in finite rings, prove a set of equivalent conditions (including $n|12$ and dihedral-product structure), and establish that if realizable in characteristic zero then $n$ must be twice an odd integer; explicit rings realize the listed finite-character cases, with several open cases remaining. The work combines group-theoretic analysis, ring-structure arguments (Jacobson radical, Artin–Wedderburn decompositions), and computational checks (e.g., SageMath/GAP) to map the landscape of realizable linear unit 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.