Result for the group SL(2,172)

Ghofran  Awad Khalaf; Niran  Sabah Jasim; Zeinab Saeidian; Azza  I.M.S. Abu-Shams

doi:10.30526/37.2.3015

Authors

Ghofran Awad Khalaf Ministry of Education, Directorate General of Education in Diyala, Diyala, Iraq https://orcid.org/0009-0005-1045-2777
Niran Sabah Jasim Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham University of Baghdad, Baghdad, Iraq https://orcid.org/0000-0001-5340-3020
Zeinab Saeidian University of Kashan, Kashan, Iran https://orcid.org/0000-0002-6279-7217
Azza I.M.S. Abu-Shams Mathematics of Department, College of Science, Philadelphia University, Amman, Jordan https://orcid.org/0009-0003-1741-1447

DOI:

https://doi.org/10.30526/37.2.3015

Abstract

The set of all (n×n) non-singular matrices over the field F this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension n over the field F, denoted by GL(n,F). The determinant of these matrices is a homomorphism from GL(n,F) into F* and the kernel of this homomorphism was the special linear group and denoted by SL(n,F). Thus SL(n,F) is the subgroup of GL(n,F) which contains all matrices of determinant one.
The rational valued characters of the rational representations written as a linear combination of the induced characters for the groups SL(2,17^2 ) discuss in this work and find the Artin indicator for this group after study the rational valued characters of the rational representations and the induced characters.

References

Mohamed, S K. On Rational -Valued Characters of Certain Types of Permutation Group. Ibn Al-Haitham journal for pure and applied sciences 2006, 19(4), 99-108.

Saad, O. B. Investigating Particular Representations for Matrix Lie Groups SO(3) and SL(2,₵). Iraqi journal of Science 2019, 60(4), 856-858. https://doi.org/10.24996/ijs.2019.60.4.19

Sigler, L.E. Algebra. Springer-Verlage, Berlin 1976.

Noor Alhuda, S. S.; Niran S. J. Periodical split for the groups PSL(2,31) and PSL(2,37). Journal of Discrete Mathematical Sciences and Cryptography 2022, 25(2), 605–608, DOI: https://doi.org/10.1080/09720529.2021.1982490

Sherouk, A. K.; Niran S. J. Calculation for the groups SL(2,U), U = 31 and 37, Journal of Discrete Mathematical,Sciences,and,Cryptography,,2022,,25(2),,609-613. https://doi.org/10.1080/09720529.2021.1972614

Mohammed I. L.; Niran S. J.; Issa A. Score for the group SL (2,38). Ibn Al-Haitham Journal for Pure and Applied Sciences 2023, 36(3), 408–415, DOI: https://doi.org/10.30526/36.3.3017

Mohammed S. A. Z.; Niran S. J.; Azza I.M.S. A.-S.; Ahmad I. Outcome for The Group SL(2,57) . Ibn Al-Haitham Journal for Pure and Applied Sciences 2024, 37(1), 403–411, DOI: https://doi.org/10.30526/37.1.3018

Salih,O. M.; Mohammed, N. J.; Alwan, B. M. Result for the Groups SUT(2,p), where p = 3, 5, 7. Technology reports of Kansai University 2020, 62(3), 2029-2034.

Dunya, M. H.; Ahmed, K. M.; Intidhar, Z. M. Score for some Groups SUT(2,p). Int. J. Nonlinear Anal. Appl. 2021, 12(2), 1-15.

Maha, A. M.; Lemya A. Al. H.; Asma A. A. New algorithm based on deep learning for number recognition. International Journal of Mathematics and Computer Science 2023, 18(3), 429–438.

Anwar, K. F.; Lemya, A. Al. H.; Areej, M. A.; Shatha A. S. Symmetric generalized bi-derivations with prime ideals. International Journal of Mathematics and Computer Science 2023, 18(4), 675–684,.

Bill, C. Representations of SL2 (R). Pacific Journal of Mathematics 2020, 3(1), 231-250.

Yurii, I. L. Introduction to the Theory of Banach Representations of Groups, Birkhäuser Verlag 2021.

Chemistry, L. Representations of Groups 2019, Kharkov, Ukraine.

Zagier, D. Applications of the representation theory of finite groups 2019, New York.

Alexander, K. Jr. An introduction to Lie groups and Lie algebras 2022, Birkhäuser Verlag.

Reiner, I. Representation theory 2021, John Wiley & Sons, NewYork –London.

Kevin, H. on representation theory in groups, 2020, Springer-Verlage.

Mohammed, S. I.K.; Lemia, A. Al. H. The Artin's Exponent of A Special Linear Group SL(2,2k ). Eng. & Tech. Journal 2010, 28(10), 1924-1934.

Yuxin, Z. T.; Orkesh, N.; Zhang, Z. Homology Algebra and Applications. J.Sci.I.R.Iran 2019, 3(5), 11-13.

Curtis, C.W.; Reiner, I. The Representation Theory Of Finite Groups And Associative Algebras 1962, John Wiley & Sons, NewYork –London.

Behravesh, H. The Rational Character Table Of Special Linear Groups. Journal of Sciences Islamic Republic Of Iran 1998, 9(2), 173-180.

Behravesh, H. Quasi – Permutation Representations Of SL(n,q) And PSL(n,q). Glasgow.Math.J. 1999, 41, 393- 408. https://doi.org/10.1017/S0017089599000567

Isaacs, I.M. Character Theory Of Finite Groups 1976, Academic Press, NewYork.

Serre, J.P. Linear Representation Of Finite Groups 1977, Springer-Verlage.

Gehles, K.E. Ordinary Characters Of Finite Special Linear Groups, M.Sc. Thesis, Dissertation, University of St Andrews, 2002.

Kirdar, M.S. The Factor Group Of The Z-Valued Class Function Modulo The Group Of The Generalized Characters, Ph.D.Thesis, University of Birmingham, 1982.

Kirdar, M.S. On Brauer’s Proof Of The Artin Induction Theorem. Abhath AL–Yarmouk (Basic Sciences and Engineering) Journal 2002, 11(1A), 51–54.

Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media, 2013.