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

  • V. M. Petechuk Transcarpathian Institute of Postgraduate Pedagogical Education
  • Yu. V. Petechuk Transcarpathian Hungarian Institute named after Ferenc Rakoczi II
Keywords: division ring, linear groups, residual and fixed modules, transvections, unipotent elements, commutators, commutativity

Abstract

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 $.

Author Biographies

V. M. Petechuk, Transcarpathian Institute of Postgraduate Pedagogical Education

Transcarpathian Institute of Postgraduate Pedagogical Education
Uzhhorod, Ukraine

Yu. V. Petechuk, Transcarpathian Hungarian Institute named after Ferenc Rakoczi II

Transcarpathian Hungarian Institute named after Ferenc Rakoczi II
Beregovo, Ukraine

References

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N.A. Vavilov, A.V. Stepanov, Linear groups over general rings. I. Generalities, Zap. Nauchn. Sem. S.-Petersburg. Otdel. Math. Inst. Steklov. (LOMI), 394 (Voprosy Teorii Predstavlenii Algebr i Grupp. 22), 295 (2011), 33–139.

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Published
2020-10-05
How to Cite
1.
Petechuk VM, Petechuk YV. Properties of the commutators of some elements of linear groups over divisions rings. Mat. Stud. [Internet]. 2020Oct.5 [cited 2020Oct.27];54(1):15-2. Available from: http://matstud.org.ua/ojs/index.php/matstud/article/view/80
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Articles