Fuchs' problem for 2-groups

Abstract: Nearly $60$ years ago, L\'{a}szl\'{o} 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 $4$ and exponent $2$ are realizable in characteristic $2$, and we prove that many $2$-groups with exponent $4$ and nilpotency class $3$ are realizable in characteristic $2$. On the other hand, we provide an example of a $2$-group with exponent $4$ and nilpotency class $4$ that is not realizable in characteristic $2$. Moreover, while some groups of exponent greater than $4$ are realizable as unit groups of rings, we prove that any $2$-group with a self-centralizing element of order $8$ or greater is never realizable in characteristic $2^m$, and consequently any indecomposable, nonabelian group with a self-centralizing element of order $8$ or greater cannot be the group of units of a finite ring.