TY - JOUR
AU - Petechuk, V. M.
AU - Petechuk, Yu. V.
PY - 2020/10/05
Y2 - 2021/12/09
TI - Properties of the commutators of some elements of linear groups over divisions rings
JF - Matematychni Studii
JA - Mat. Stud.
VL - 54
IS - 1
SE - Articles
DO - 10.30970/ms.54.1.15-22
UR - http://matstud.org.ua/ojs/index.php/matstud/article/view/80
SP - 15-22
AB - Inclusions resulting from the commutativity of elements and their commutators with trans\-vections in the language of residual and fixed submodules are found.The residual and fixed submodules of an element $\sigma $ of the complete linear group are defined as the image and the kernel of the element $\sigma -1$ and are denoted by $R(\sigma )$ and $P(\sigma )$, 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 $\tau $ from the commutativity of the commutator $\left[g,\tau \right]$ with $g$ is followed by the inclusion of $R(\left[g,\tau \right])\subseteq P(\tau )\cap P(g)$. It is proved that the same inclusions occur over an arbitrary division ring if $g$ is a unipotent element, $\mathrm{dim}\mathrm{}(R\left(\tau \right)+R\left(g\right))\le 2$ and the commutator $\left[g,\tau \right]$ commutes with $\tau $ or if $g$ is a unipotent commutator of some element of the complete linear group and transvection $\ \tau $.
ER -