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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Ivan Franko National University of Lviv
2020-10-01
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| Series: | Математичні Студії |
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/80 |
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| author | V. M. Petechuk Yu. V. Petechuk |
| author_facet | V. M. Petechuk Yu. V. Petechuk |
| author_sort | V. M. Petechuk |
| collection | DOAJ |
| description | 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 $. |
| format | Article |
| id | doaj-art-45e8936f0d2b4a8da8616de2ea255491 |
| institution | DOAJ |
| issn | 1027-4634 2411-0620 |
| language | deu |
| publishDate | 2020-10-01 |
| publisher | Ivan Franko National University of Lviv |
| record_format | Article |
| series | Математичні Студії |
| spelling | doaj-art-45e8936f0d2b4a8da8616de2ea2554912025-08-20T02:41:33ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202020-10-01541152210.30970/ms.54.1.15-2280Properties of the commutators of some elements of linear groups over divisions ringsV. M. Petechuk0Yu. V. Petechuk1Transcarpathian Institute of Postgraduate Pedagogical EducationTranscarpathian Hungarian Institute named after Ferenc Rakoczi IIInclusions 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 $.http://matstud.org.ua/ojs/index.php/matstud/article/view/80division ringlinear groupsresidual and fixed modulestransvectionsunipotent elementscommutatorscommutativity |
| spellingShingle | V. M. Petechuk Yu. V. Petechuk Properties of the commutators of some elements of linear groups over divisions rings Математичні Студії division ring linear groups residual and fixed modules transvections unipotent elements commutators commutativity |
| title | Properties of the commutators of some elements of linear groups over divisions rings |
| title_full | Properties of the commutators of some elements of linear groups over divisions rings |
| title_fullStr | Properties of the commutators of some elements of linear groups over divisions rings |
| title_full_unstemmed | Properties of the commutators of some elements of linear groups over divisions rings |
| title_short | Properties of the commutators of some elements of linear groups over divisions rings |
| title_sort | properties of the commutators of some elements of linear groups over divisions rings |
| topic | division ring linear groups residual and fixed modules transvections unipotent elements commutators commutativity |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/80 |
| work_keys_str_mv | AT vmpetechuk propertiesofthecommutatorsofsomeelementsoflineargroupsoverdivisionsrings AT yuvpetechuk propertiesofthecommutatorsofsomeelementsoflineargroupsoverdivisionsrings |