International Journal of Nonlinear Analysis and Applications

On subgroups of the unitary group especially of degree 2

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Mathematics and natural sciences, University of Brawijaya, Malang, Indonesia

2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia

Abstract

The point of the current investigation is to research one of the extremely significant groups exceedingly associated with the classical group which is called the special unitary groups $SU_{2}(K)$ particularly of degree $2$. Let $K$ be a field of characteristic, not equal $2$, our principal objective that to depicting subgroups of $SU_{2}(K)$ over a field $K$ contains all elementary unitary transvections.

Keywords

[1] A. Bak and N. Vavilov, Structure of hyperbolic unitary groups, I. Elementary subgroups, Algebra Colloq. 7(2) (2000) 159–196.
[2] A. Bak and N. Vavilov, The normal structure of hyperbolic unitary groups, Dissertation PhD, University of Bielefeld, Germany, DIN-ISO 9706, 2014.
[3] E.L. Bashkirov, On subgroups of the special linear group of degree 2 over an infinite field, Sbornik Math. 187 (1996) 175–192.
[4] E.L. Bashkirov, Linear groups that contain a root subgroup, Sib. Math. J. 37 (1996) 754-759.
[5] E.L. Bashkirov, On subgroups of the general linear group over the skew field of quaternions containing the special unitary group, (Russian.) Sibirsk. Mat. Zh. 39 (1998) 1251–1266.
[6] E.L. Bashkirov, On subgroups of the general linear group of degree four over the skew field of quaternions containing the special unitary group of maximal Witt index, Commun. Alg. 42 (2014) 704–728.
[7] E.L. Bashkirov, On subgroups of the group GL7 over a field that contains a Chevalley group of type G2 over a subfield, J. Pure Appl. Alg. 219 (2015) 1992-2014.
[8] E.L. Bashkirov, On linear group of degree 2 over a noncommutative division algebra, Commun. Alg. 44(2016) 1033–1054.
[9] E.L. Bashkirov and C.K. Gupta, Linear groups over locally finite extensions of infinite fields. Int. J. Alg. Comp. 17 (2007) 905–922.
[10] E.L. Bashkirov and C.K. Gupta, Linear groups over integral extensions of semilocal commutative rings, Commun. Alg. 42 (2014) 4149–4171.
[11] L.E. Dickson, Linear Groups with an Exposition of Galois Field Theory, Dover, New York, 2003.
[12] J. Dieudonne, On the structure of unitary groups, Trans. Amer. Math. Soc.72 (1952) 367–385.
[13] R.H. Dye, The conjugacy classes of fixed point free elements of order three or four in GL2n(K), SL2n(K), P GL2n(K), and PSL2n(K), J. Algebra, 169 (1994), 399–417.
[14] E. W. Ellers, J. Malzahn, The length problem for the special unitary group generated by positive transvections, J. Univ. Kuwait Sci. 20 (1993) 1–24.
[15] C.L. Grove, Classical Groups and Geometric Algebra, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
[16] O.H. King, On subgroups of the special linear group containing the special unitary group, Geometric Dedicate 19 (1985) 297-310.
[17] O.H. King, On subgroups of the special linear group containing the special orthogonal group, J. Algebra 96 (1985) 178-193.
[18] S.M. Sabbar, Certain subgroups of the projective special linear P SL2(K) over a field K, Master Dissertation, Fatih University, Istanbul, Turkey, 2015.
[19] A.J. Rotman, An Introduction to the Theory of Groups, Fourth edition, Graduate Texts in Mathematics, 148. Springer-Verlag, New York, 1995.
[20] S.M. Sabbar, Description of the projective transvection of the projective special linear group PSL2(K) Over a Field K, J. Phys.: Conf. Ser. 1562 (2020) 012012.
[21] S.M. Sabbar, Subgroups of the projective special linear group P SL2(K) that contain a projective root subgroup, Appl. Sci. 22 (2020) 205-215.
[22] L. Shangzhi, Over group of SU(n, K, f) or Ω(n, K, Q) in GL(n, K), Geom. Ded. 33 (1990) 241-250.
[23] Li. Shangzhi, Overgroups of a unitary group in GL(2, K), J. Algebra  149 (1992) 275-286.
[24] A.E. Zalesskii and V.N. Serezhkin, Linear groups generated by transvections, Izv. Akad. Nauk SSSR. Ser. Mat. 40 (1976) 26-49.
International Journal of Nonlinear Analysis and Applications
Volume 12, Issue 1
May 2021
Pages 1115-1121
  • Receive Date: 02 February 2021
  • Revise Date: 06 April 2021
  • Accept Date: 11 April 2021