TY - JOUR

T1 - Bijections preserving commutators and automorphisms of unitriangular group

AU - Holubowski, W.

AU - Stepanov, A.

PY - 2017

Y1 - 2017

N2 - We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $\UT(n,F)$ over a field $F$, where $n\in\N\cup\{\infty\}$. PC-maps were recently described up to almost identity PC-maps by D.Wang, S.Ou, and W.Zhang for finite $n$ and by R.Slowik 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 $\UT(\infty, F)\to\UT(\infty, F)$ is an autmorphism.

AB - We complete characterization of bijections preserving commutators (PC-maps) in the group of unitriangular matrices $\UT(n,F)$ over a field $F$, where $n\in\N\cup\{\infty\}$. PC-maps were recently described up to almost identity PC-maps by D.Wang, S.Ou, and W.Zhang for finite $n$ and by R.Slowik 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 $\UT(\infty, F)\to\UT(\infty, F)$ is an autmorphism.

U2 - 10.1080/03081087.2016.1165170

DO - 10.1080/03081087.2016.1165170

M3 - Article

VL - 65

SP - 23

EP - 34

JO - Linear and Multilinear Algebra

JF - Linear and Multilinear Algebra

SN - 0308-1087

IS - 1

ER -