Subgroups of the skew general linear group over the division ring of real quaternions
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Abstract
Let H be the division ring of real quaternions and n a positive integer. In this paper, we show that every subgroup of finite index in the skew general linear group GL(n,H) is non-central normal and so it contains the skew special linear group SL(n,H). Also, we show that every proper subgroup of SL(n,H) is of infinite index. Besides, we show that every subnormal subgroup of GL(n,H) is a T-group and so it is normal in GL(n,H).
Keywords
division ring of real quaternions, subgroup of finite index, subnormal subgroup
Article Details
References
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2. Lê Văn Chua, Nhóm con của nhóm nhân trong vành chia quaternion thực, Ho Chi Minh City University of education Journal of Science, Tập 16, Số 12 (2019): 975-981.
3. P. K. Draxl, Skew Fields, London Math. Soc. Lecture Note Series, vol. 81, Cambridge University Press, 1983.
4. G. R. Greenfield, A note on subnormal subgroups of division algebras. Can. J. Math, 30 (1978), 161-163.
5. I. Kaplansky, A theorem on division rings, Can. J. Math, 3 (1951), 290-292.
6. I. N. Herstein and W. R. Scott, Subnormal subgroups of division rings. Can. J. Math, 15 (1963), 80-83.
7. L. K. Hua, On the multiplicative group of a field, Acad. Sinic. Sci. Record 3 (1950), 1-6
8. I. Schur, Uber die darstellung der endlichen gruppen durch gebrochene lineare substitutionen, J. Reine Angew. Math. 127 (1904), 20-50.
9. M. Shirvani and B. A. F. Wehrfritz, Skew Linear Groups, Cambridge University Press, 1986.
10. A. E. Zalesskii, Solvable groups and crossed products, Math. Sb. (N.S) 67 (109), 1965, 154-160.