Bijections preserving commutators and automorphisms of unitriangular group

TL;DR

This paper fully characterizes bijections that preserve commutators in the group of unitriangular matrices over a field, showing that such maps are essentially automorphisms, especially in the infinite case.

Contribution

It completes the classification of PC-maps in unitriangular groups, demonstrating that almost identity PC-maps are central multiplications or identities, leading to automorphism results.

Findings

Almost identity PC-maps are central multiplications.

In the infinite case, almost identity PC-maps are identities.

All PC-maps of $UT( ext{infinity},F)$ are automorphisms.

Abstract

We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $UT(n,F)$ over a field $F$, where $n$ is a natural number or infinity. PC-maps were recently described up to almost identity PC-maps by M.Chen, D.Wang, and H.Zhai (2011) for finite $n$ and by R.Slowik (2013) for $n=\infty$. An almost identity map is a map, preserving elementary transvections. We show that an almost identity PC-map is a multiplication by a central element. In particular, if $n=\infty$, then an almost identity map is identity. Together with the result of R.Slowik this shows that any PC-map of $UT(\infty, F)$ is an automorphism.