TORSION MAXIMAL SUBGROUPS OF GLn(D)

Document Type : Original Manuscript

Author

Department of Mathematics Education, Farhangian University, P.O. Box 14665-889, Tehran, Iran.

10.22044/jas.2025.14465.1831

Abstract

Let D be a division ring over its center F. Let
GLn(D)
be the general linear group over D. In this article we prove that
if
GLn(D)
contains a maximal torsion subgroup M then D = F,
Fp
charF = p > 0 and F is algebraic over its prime sub eld
when-
ever one of the following conditions occurs: (1) D is algebraic over
F; (2) there exists an element a
2
M such that
CMn(D)(F[a])
is
algebraic over F.

Keywords

References

1. S. Akbari, R. Ebrahimian, H. Momenaee Kermani and A. Salehi Golsefidy, Maximal subgroups of GLn(D), J. Algebra, 259 (2003), 201–225.
2. S. Akbari and M. Mahdavi-Hezavehi, Some special subgroup of GLn(D), Algebra Colloq., 5(4) (1998), 361–370. 
3. S. Akbari, M. Mahdavi-Hezavehi and M. G. Mahmudi, Maximal subgroups of GL1(D), J. Algebra, 217 (1999), 422–433.
4. E. Artin, Geometric Algebra, Interscience Publications, New York, 1957.
5. H. Dorbidi, R. Fallah-Moghaddam and M. Mahdavi-Hezavehi, Existence of non-abelian free subgroups in the maximal subgroups of GLn(D), Algebra Colloq., 21(3) (2014), 483–496.
6. H. Dorbidi, R. Fallah-Moghaddam and M. Mahdavi-Hezavehi, Soluble maximal subgroups of GLn(D), J. Algebra Appl., 9(6) (2010), 921–932.
7. P. K. Draxl, Skew fields, Cambridge Univ. Press, Cambridge, 1983.
8. R. Fallah-Moghaddam and M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of GLn(D), Comm. Algebra, 45 (2017), 3724–3729.
9. R. Fallah-Moghaddam and M. Mahdavi-Hezavehi, Locally finite conditions on maximal subgroups of GLn(D), Algebra Colloq., 19(1) (2012), 73–86.
10. J. Z. Goncalves, Free groups in subnormal subgroups and the residual nilpotence of the group of units of group rings, Canad. Math. Bull., 27 (1984), 365–370.
11. J. Z. Goncalves and A. Mandel, Are there free groups in division rings?, Israel J. Math., 51(1) (1986), 69–80.
12. R. Hazrat, M. Mahdavi-Hezavehi and M. Motiee, Multiplicative groups of division rings, Math. Proc. R. Ir. Acad., 114(1) (2014), 37–114.
13. R. Hazrat and A. R. Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, J. Algebra, 322 (2009), 2528–2543.
14. I. N. Herstein, Multiplicative commutators in division rings, Israel J. Math., 31(2) (1978), 180–188.
15. L. K. Hua, On the multiplicative group of a sfield, Acad. Sinic. Sci. Record, (1950), 1–6.
16. T. Y. Lam, A first course in non-commutative rings, GTM 131, Springer-Verlag, Berlin, 2001.
17. S. Li, The maximality of monomial subgroups of linear groups over division rings, J. Algebra, 127(1) (1989), 22–39.
18. A. E. Lichtman, On subgroups of the multiplicative group of skew fields, Proc. Amer. Math. Soc., 63 (1977), 15–16.
19. M. Mahdavi-Hezavehi, Free subgroups in maximal subgroups of GL1(D), J. Algebra, 241(2) (2001), 720–730.
20. M. Ramezan-Nassab and D. Kiani, Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups, J. Algebra, 399 (2014), 269–276.
21. A. Rosenberg, The Cartan-Bruer-Hua theorem for matrix and local matrix rings, Proc. Amer. Math. Soc., 7 (1956), 891–898.
22. W. R. Scott, Group theory, Dover, New York, 1987.
23. M. Shirvani and B. A. F. Wehrfritz, Skew linear groups, London Math. Soc. Lecture Note Ser., Vol. 118, 1986.
24. D. A. Suprunenko, Matrix groups, Amer. Math. Soc. Providence, RI, 1976.
25. J. Tits, Free subgroups in linear groups, J. Algebra, 20 (1972), 250–270. 
Journal of Algebraic Systems
Volume 14, Issue 3
June 2026
Pages 477-488

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