# 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.

### References

M. Aaghabali, M. H. Bien, Subnormal subgroups and self-invariant maximal subfields in division rings, J. Algebra, 586, 844–856 (2021). DOI: https://doi.org/10.1016/j.jalgebra.2021.07.014

D. Z. Dokovic, On commutators in real semisimple Lie groups, Osaka J. Math., 23, 223–228 (1986).

M. Droste, I. Rivin, On extension of coverings, Bull. London Math. Soc., 42, 1044–1054 (2010). DOI: https://doi.org/10.1112/blms/bdq068

E. A. Egorchenkova, N. L. Gordeev, Products of commutators on a general linear group over a division algebra, J. Math. Sci., 243, 561–572 (2019). DOI: https://doi.org/10.1007/s10958-019-04556-8

E. W. Ellers, N. Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc., 350, 3657–3671 (1998). DOI: https://doi.org/10.1090/S0002-9947-98-01953-9

A. Kanel-Belov, B. Kunyavskii, E. Plotkin, Word equations in simple groups and polynomial equations in simple algebras, Vestnik St. Petersburg Univ. Math., 46, 3–13 (2013). DOI: https://doi.org/10.3103/S1063454113010044

V. V Kursov, The commutant of the general linear group over a field, Dokl. Akad. Nauk BSSR, 23, 869–871 (1979).

C. Y. Hui, M. Larsen, A. Shalev, The Waring problem for Lie groups and Chevalley groups, Israel J. Math., 210, 81–100 (2015). DOI: https://doi.org/10.1007/s11856-015-1246-9

T. Y. Lam, {A first course in noncommutative rings, 2nd ed., Grad. Texts in Math., 131, Springer-Verlag, New York (2001). DOI: https://doi.org/10.1007/978-1-4419-8616-0

M. Mahdavi-Hezavehi, M. Motiee, Division algebras with radicable multiplicative groups, Commun. Algebra, 39, 4084–4096 (2011). DOI: https://doi.org/10.1080/00927872.2010.517819

M. Mahdavi-Hezavehi, Commutators in division rings revisited, Bull. Iranian Math. Soc., 26, 7–88 (2000).

T. Nakayama, Y. Matsushima, Über die multiplikative Gruppe einer p-adischen Divisionsalgebra, Proc. Imperial Acad. (Tokyo), 19, 622–628 (1943). DOI: https://doi.org/10.3792/pia/1195573246

O. Ore, Some remarks on commutators, Proc. Amer. Math. Soc., 2, 307–314 (1951). DOI: https://doi.org/10.1090/S0002-9939-1951-0040298-4

R. Ree, Commutators in semi-simple algebraic groups, Proc. Amer. Math. Soc., 15, 457–460 (1964). DOI: https://doi.org/10.1090/S0002-9939-1964-0161944-X

D. Segal, {Words: notes on verbal width in groups, London Math. Soc. Lecture Notes Ser., 361, Cambridge Univ. Press, Cambridge (2009). DOI: https://doi.org/10.1017/CBO9781139107082

K. Shoda, Einige Sätze über Matrizen, Japan J. Math., 13, 361–365 (1937). DOI: https://doi.org/10.4099/jjm1924.13.0_361

A. Thom, A. Elkasapy, About Got^os method showing surjectivity of word maps, Indiana Univ. Math. J., 63, 1553–1565 (2014). DOI: https://doi.org/10.1512/iumj.2014.63.5391

R. C. Thompson, Commutators in the special and general linear groups, Trans. Amer. Math. Soc., 101, 16–33 (1961). DOI: https://doi.org/10.1090/S0002-9947-1961-0130917-7

L. N. Vaserstein, E. Wheland, Commutators and companion matrices over rings of stable rank $1$, Linear Algebra and Appl., 142, 263–277 (1990). DOI: https://doi.org/10.1016/0024-3795(90)90270-M

Published
11.04.2023
How to Cite
Bien, M. H., P. L. P. Lam, and V. T. Mai. “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.
Issue
Section
Research articles