Properties of the commutators of some elements of linear groups over divisions rings
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 elem...
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| Main Authors: | , |
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| Format: | Article |
| Language: | deu |
| Published: |
Ivan Franko National University of Lviv
2020-10-01
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| Series: | Математичні Студії |
| Subjects: | |
| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/80 |
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| Summary: | 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 $. |
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| ISSN: | 1027-4634 2411-0620 |