Properties of the commutators of some elements of linear groups over divisions rings

Abstract

Inclusions resulting from the commutativity of elements and their commutators with transvections in the language of residual and fixed submodules and found. The residual and fixed submodules of an element σ of the complete linear group are defined as the image and the kernel of the element σ−1 and are denoted by R(σ) and P(σ), respectively. It is shown that for an arbitrary element g of a complete linear group over a division ring whose characteristic is different from 2 and the transvection τ from the commutativity of the commutator [g, τ ] with g is followed by inclusion of R([g, τ ]) ⊆ P(τ ) ∩ P(g).It is proved that the same inclusions occur over an arbitrary division ring if g is a unipotent element,dim(R (τ ) + R (g)) ≤ 2 and the commutator [g, τ ] commutes with τ or if g is a unipotent commutator of some element of the complete linear group and transvection τ.