Which finite simple groups are unit groups?

Abstract

Abstract. We prove that if G is a finite simple group which is the unit group of a ring, then G is isomorphic to either (a) a cyclic group of order 2; (b) a cyclic group of prime order 2 k − 1 for some k; or (c) a projective special linear group PSLn(F 2 ) for some n ≥ 3. Moreover, these groups do (trivially) all occur as unit groups. We deduce this classification from a more general result, which holds for groups G with no non-trivial normal 2-subgroup.

Fuchs' problem for 2-groups

“…We begin with some examples that we find interesting, and we end with some questions we would like to see answered. For a nonabelian example, we can take M 16 , the Modular or Isanowa group of order 16, which is group SmallGroup (16,6) in GAP. This group has presentation…”

“…Next, we collect some elementary, but extremely useful, observations about finite rings and their unit groups in characteristic m. As noted in [6,Lem. 6] and [2, Prop.…”

“…All finite realizable groups of odd order were described by Ditor in [9]. In the past decade, Davis and Occhipinti determined all realizable finite simple groups [6], as well as all realizable alternating and symmetric groups [5]. During the same time period, Chebolu and Lockridge solved Fuchs' problem for dihedral groups [2].…”

Finitely generated abelian groups of units

“…Much has been said about the units of group rings. Recently, the finite dihedral groups and the simple groups that are realizable as the group of units of a ring have been classified (see and ).…”

Fuchs' problem for dihedral groups

“…Call a group G realizable if it is the group of units of some ring. In [DO14a] and [DO14b] the authors determine which alternating, symmetric, and finite simple groups are realizable. In the present work, we determine which dihedral groups are the group of units of a ring, and our classification is stratified by characteristic.…”