Bijective maps preserving commutators on a solvable classical group.
type: Item
instance of: scholarly article
title: Bijective maps preserving commutators on a solvable classical group.
zbMATH Open document ID: 1223.20022
author: Dengyin Wang
author: Shikun Ou
author: Wei Zhang
published in: Science China. Mathematics
publication date: 2011-02-25T00:00:00Z
review text: Let \(G\) be a group. A bijective map \(\varphi\colon G\to G\) is called a PC-map (means preserving commutators) of \(G\) if \(\varphi([x,y])=[\varphi(x),\varphi(y)]\) for all \(x,y\in G\). In the paper, the authors study these maps on a group. Obviously, an automorphism of \(G\) is a PC-map. In the present paper, the authors give some examples to show that the converse is false. More exactly, these examples are constructed for the matrix group \(\mathcal U\) of all \(n\times n\) unit upper triangular matrices over a field \(F\). Further, the authors study PC-maps on the group \(\mathcal T_n\) of all \(n\times n\) invertible upper triangular matrices. The main theorem of the paper gives necessary and sufficient conditions for a map \(\varphi\colon\mathcal T_n\to\mathcal T_n\) to be a PC-map.
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 15A86
Mathematics Subject Classification ID: 20H20
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 5857635
zbMATH Keywords: linear preserver maps
zbMATH Keywords: maps preserving commutators
zbMATH Keywords: matrix groups
Wikidata QID: Q114222428
reviewed by: Bùi Xuân Hải
MaRDI profile type: Publication
full work available at URL: s11425-010-3133-5
OpenAlex ID: W2013494467
cites work: Willmore tori with umbilic lines and minimal surfaces in hyperbolic space
cites work: Non-Linear Spectrum-Preserving Maps
cites work: Linear Preserver Problems
cites work: Linear preserver problems: A brief introduction and some special techniques
cites work: Linear Operations on Matrices
cites work: Nonlinear maps preserving solvability
cites work: Q5478096
DOI: 10.1007/S11425-010-3133-5
description: scientific article; zbMATH DE number 5857635