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 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.

中文翻译：

---

2 组的 Fuchs 问题

大约 60 美元前，Laszlo Fuchs 提出了确定哪些群可以实现为交换环的单位群的问题。迄今为止，这个问题仍然悬而未决，尽管已经取得了重大进展。沿着这条路线，人们还可以提出更一般的问题，即哪些有限群可以实现为有限环的单位群。在本文中，我们考虑哪些$2$-群可实现为有限环的单位群的问题，这是确定哪些幂零群可实现的必要步骤。我们证明指数$4$的所有$2$-群在特征$2$中是可实现的。此外，虽然一些大于 $4$ 的指数组可以作为环的单位组实现，但我们证明任何具有 $8$ 或更高阶自集中元素的 $2$-组永远不会在特征 $2^m$ 中实现，