Bijective maps preserving commutators on a solvable classical group.
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.
scientific article; zbMATH DE number 5857635