Bijections preserving commutators and automorphisms of unitriangular groups

Abstract

We complete the characterization of bijection preserving commutators (PC-maps) on the group of unitriangular matrices over a field F, where . PC-maps were recently described up to almost identity PC-maps by Chen et al.[18] for finite n and by Słowik[17] for . An almost identity map is a map fixing elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if , then an almost identity map is identity. In view of the result of Słowik, this shows that any PC-map of is an automorphism.