Journal of Mathematical Sciences

, Volume 243, Issue 4, pp 561–572 | Cite as

Products of Commutators on a General Linear Group Over a Division Algebra

  • E. A. EgorchenkovaEmail author
  • N. L. GordeevEmail author

The word maps \( \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) \) and \( \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } \) for a word \( w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], \) where D is a division algebra over a field K, are considered. It is proved that if \( \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], \) then \( \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), \) where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then \( \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). \) The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Russian State Pedagogical UniversitySt.PetersburgRussia

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