Commutators in special linear groups over certain division rings

  • M. H. Bien Faculty of Mathematics and Computer Science, University of Science and Vietnam National University, Ho Chi Minh City, Vietnam
  • P. L. P. Lam Faculty of Mathematics and Computer Science, University of Science and Vietnam National University, Ho Chi Minh City, Vietnam
  • V. T. Mai Faculty of Mathematics and Computer Science, University of Science and Vietnam National University, Ho Chi Minh City, Vietnam
Keywords: Commutator; Derived word; Derived series; Mal'cev-Neumann Division Ring

Abstract

UDC 512.5

We consider the question whether an element of a special linear group  ${\rm SL}_m(D)$ of degree $m\ge 1$ over a division ring $D$  is a commutator. Our first aim  is to show that if the division ring $D$ is algebraically closed and finite-dimensional over its center, then every element of ${\rm SL}_m(D)$ is a commutator of ${\rm SL}_m(D).$  We also indicate that this question  is related to the derived series in division rings and then describe the derived series in the Mal'cev–Neumann division rings of noncyclic free groups over fields. 

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Published
11.04.2023
How to Cite
BienM. H., LamP. L. P., and MaiV. T. “Commutators in Special Linear Groups over Certain Division Rings”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 3, Apr. 2023, pp. 328 -36, doi:10.37863/umzh.v75i3.6872.
Section
Research articles