Fuchs' problem for 2-groups

Nearly 60 years ago, Laszlo Fuchs posed the problem of determining which groups can be realized as the group of units of a commutative ring. To date, the question remains open, although significant progress has been made. Along this line, one could also ask the more general question as to which finite groups can be realized as the group of units of a finite ring. In this paper, we consider the question of which 2-groups are realizable as unit groups of finite rings, a necessary step toward determining which nilpotent groups are realizable. We prove that all 2-groups of exponent 4 are realizable in characteristic 2. Moreover, while some groups of exponent greater than 4 are realizable as unit groups of rings, we establish general constraints on the exponent of a realizable 2-group. These constraints are used to describe examples of 2-groups that cannot be the group of units of a finite ring. (C) 2020 Elsevier Inc. All rights reserved.