Fuchs' problem for 2-groups (vol 556, pg 225, 2020)

In [7, Theorem 1.1], it is claimed that every finite 2-group of exponent 4 occurs as the group of units of a finite ring with characteristic 2. We now know this claim to be false: specifically, [7, Proposition 3.9] and its proof are incorrect. The purpose of this note is to provide a revised statement of [7, Theorem 1.1], which holds for any finite 2-group of exponent 4 and nilpotency class at most 2 and for some (perhaps most) finite 2-groups of exponent 4 and nilpotency class 3. We also exhibit an example of a group of order 64 with exponent 4 and nilpotency class 4 that is not realizable in characteristic 2. (C) 2022 Elsevier Inc. All rights reserved.

Automated summary

This article points out the incorrect claim made in [7, Theorem 1.1] that every finite 2-group can serve as the group of units of a finite ring with characteristic 2. A revised statement is provided, which holds for any finite 2-group of exponent 4 and nilpotency class at most 2, and for some finite 2-groups of exponent 4 and nilpotency class 3. Additionally, an example is presented of a group of order 64 with exponent 4 and nilpotency class 4 that cannot be realized in characteristic 2.