IHJPAS. 36 (3) 2023 408 This work is licensed under a Creative Commons Attribution 4.0 International License * Corresponding Author: niraan.s.j@ihcoedu.uobaghdad.edu.iq Abstract The set of all (n×n) non-singular matrices over the field F. And this set forms a group under the operation of matrix multiplication. This group is called the general linear group of dimension over the field F, denoted by . The determinant of these matrices is a homomorphism from into F* and the kernel of this homomorphism was the special linear group and denoted by Thus is the subgroup of which contains all matrices of determinant one. The rational valued characters of the rational representations are written as a linear combination of the induced characters for the groups discussed in this paper. We find the Artin indicator for this group after studying the rational valued characters of the rational representations and the induced characters. Keywords: Rational character table, Induced characters table, Artin indicator. 1. Introduction The group of all matrices of determinant 1 is (n,F), [1 and 2], and researchers in [3] defined the representation of the group. Authors in [4] survey and got the calculation for the groups SL(2, U), U = 31 and 37. We apply the same idea in [4] to compute the character table of rational representations for the group. Also, we apply the same idea in [5] to compute the Artin indicator for this group. In this work, we count all cyclic subgroups, the Artin indicator, the rationally valued characters of the rational representations, and the induced characters for the group. doi.org/10.30526/36.3.3017 Article history: Received 2022, Accepted November 2022, Published in July 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Score for the group 𝓢𝓛(2,3 8 ) Mohammed Ibrahem Lfta Department of Mathematics, Ministry of Education, Directorate General of Education karkh 3, Baghdad, Iraq. mohammed.ibrahim1203a@ihcoedu.uobaghdad.e Niran Sabah Jasim * Department of Mathematics, College of Education for Pure Science, Ibn al-Haytham, Universityof Baghdad, Baghdad, Iraq. niraan.s.j@ihcoedu.uobaghdad.edu.iq Ahmad Issa Karabük University, Faculty of Science, Department of Mathematics, Karabük, Türkiye ahmad93.issa18@gmail.com mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com http://en.uobaghdad.edu.iq/?page_id=15060 http://en.uobaghdad.edu.iq/?page_id=15060 mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com mailto:mohammed.ibrahim1203a@ihcoedu.uobaghdad.edu.iq mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq mailto:ahmad93.issa18@gmail.com IHJPAS. 36 (3) 2023 409 2. Basic Definitions and Facts Theorem 2.1: [1] = – . Definition 2.4: [1] Let H be a cyclic subgroup of group G, and  be a class function of H. Then m GG i i 1H C (g) (g) (x ) C (g)     Definition 2.5: [6] The character induced from the principal character of cyclic subgroups of G is called Artin character. Definition 2.6: [6] Let G be a finite group and let  be any rational valued character on G. The smallest positive number n such that, c c c n a   Where acZ and c is Artin character called the Artin exponent of G and is denoted by A (G). 3. The Results Authors in [7-10] studied the character table of rational representations for the group we apply that idea and compute the character table of rational representations for the group . Also we use the same idea in [5] to compute the Artin indicator for this group. The character table of rational representations for the group is: IHJPAS. 36 (3) 2023 410 This group has 32 cyclic subgroups generated by the conjugacy classes of the group. The induced character table for this group is: IHJPAS. 36 (3) 2023 411 Hence, the rational valued characters in the first table are written as a linear combination of induced characters in the second table. 1 = + + + + + + + + + + + + + + + + + + + + + + + + + + + - , Ψ = - + + + + + + + + + + + + + + + + + + + + + + + , = 624.39024 + 1248.78049 512 + 1024 + 1280 1280 + 2560 + + , = 249.75609 156.09756 156.09756 + 312.195122 128 + 400 + 160 320 + + , IHJPAS. 36 (3) 2023 412 = 62.43902 +78.04878 124.87805 78.04878 78.04878 + + 160 64 + 100 + 200 + 80 160 +160 , = 124.87805 + 156.0956 + 78.04871 + 256 160 512 + 320 320 640 + , = 15.60976 + 19.51219 + 31.21951 + 39.02439 62.43904 39.02439 39.02439 40 + 78.04878 + 80 32 + 50 + 64 200 + 40 -320 + 80 + , = 15.60976 + 19.51219 + 31.21951 + 39.02439 62.43904 78.04878 62.43902 + 64 40 128 + 80 + 80 + , = 3.90244 + 4.87805 + 7.80488 + 9.75609 + 15.60976 + 19.51219 31.21951 19.51219 20 19.51219 20 + 39.02439 + 40 16 + 25 + 32 50 + 20 40 + 40 + , = 3.90244 + 4.87805 + 7.80488 + 9.75609 + 15.60976 + 19.51219 - 31.21951 39.02439 40 + 50 32 20 64 + 40 + 40 , = 3.90244 + 4.87805 + 7.80488 + 9.75609 + 15.60976 + 19.51219 + 31.21951 + 39.02439 + 40 -39.02439 40 39.02439 40 + 100 + 80 32 + 50 100 + 40 + , = 0.97561 + 1.21951 + 1.95122 + 2.43902 + 3.90244 + 4.87805 +7.80488 + 9.75609 + 10 19.5119 20 -12.5 + 25 + 16 10 + 20 + 20 , = 1.24878 + 1.56098 + 2.49756 + 3.12195 + 4.99512 + 6.24390 + 9.99024 + 12.4878 +12.8 24.97561 25.6 16 + 32 32 + 64 + + , = 0.24390 + 0.30488 + 0.48780 + 0.60976 + 0.97561 + 1.21951 +1.95129 + 2.43902 + 2.5 + 4.87804 + 5 9.75609 IHJPAS. 36 (3) 2023 413 10 6.25 6.25 + 12.5 5 +10 + 10 , = 0.15609 + 0.19512 + 0.31219 + 0.39024 + 0.62439 + 0.78049 + 1.24878 + 1.56098 + 1.6 + 3.12195 + 3.2 6.24390 6.4 4 4 + 8 + 8 – , = 0.24390 + 0.30488 + 0.48780 + 0.60976 + 0.97561 + 1.21951 + 1.95129 + 2.43902 + 2.5 + 4.87804 +5 + 9.75609 + 10 + 12.5 20 12.5 12.55 + 25 10 + 10 + – , = 0.03902 + 0.04878 + 0.07805 + 0.09756 + 0.15609 + 0.19512 + 0.31295 + 0.39024 + 0.4 + 0.78049 + 0.8 + 1.56098 + 1.6 + 2 3.2 -2 2 + 4 4 , = 0.07804 + 0.097561 + 0.15609 + 0.19512 + 0.31219 + 0.39024 + 0.62439 + 0.78049 + 0.8 + 1.56098 + 1.6 + 3.12195 + 3.2 + 4 6.4 8 6.4 + 12.8 8 + 16 - , = 0.00976 + 0.01219 + 0.01951 + 0.02439 + 0.03902 + 0.04878 + 0.07805 + 0.09756 + 0.1 + 0.19512 + 0.2 + 0.39024 + 0.4 + 0.5 + 0.78049 + 1.6 - + 2 + 2 - , = 0.00976 + 0.01219 + 0.01951 + 0.02439 + 0.03902 + 0.04878 + 0.07805 + 0.09756 + 0.1 + 0.19512 + 0.2 + 0.39024 + 0.4 + 0.5 + 0.78049 + 1.6 2 1.6 + 2 , = 0.00244 + 0.00305 + 0.00488 + 0.00609 + 0.00976 + 0.01219 + 0.01951 + 0.02439 + 0.025 + 0.04878 + 0.05 + 0.09756 + 0.1 + 0.125 + 0.2 + 0.25 0.4 + 0.5 + 0.8 0.5 0.5 + + , = 0.00244 + 0.00305 + 0.00488 + 0.00609 + 0.00976 + 0.01219 + 0.01951 + 0.02439 + 0.025 + 0.04878 + 0.05 + 0.09756 + 0.1 IHJPAS. 36 (3) 2023 414 + 0.125 + 0.2 + 0.25 0.4 + 0.5 0.8 + - , = 0.00244 + 0.00305 + 0.00488 + 0.00609 + 0.00976 + 0.01219 + 0.01951 + 0.02439 + 0.025 + 0.04878 + 0.05 + 0.09756 + 0.1 + 0.125 + 0.2 + 0.25 -0.4 + 0.5 + 0.8 + - - + - , = 0.00061 + 0.00076 + 0.00122 + 0.00152 + 0.00244 + 0.00305 + 0.00488 + 0.00609 + 0.00625 + 0.01219 + 0.0125 + 0.02439 + 0.025 + 0.03125 + 0.05 + 0.0625 + 0.1 + 0.125 + 0.2 + 0.25 0.5 + 0.5 – , = 764.02073 1528.04145 + 722.82353 3510.85714 1536 + 1536 + 1536 3072 + – , = 764.02073 + 722.82353 + 1755.42857 768 768 + , = 47.75129 + 95.50259 219.42857 + 96 96 + 96 192 + + , = 23.87565 47.75129 135.52941 + 96 96 48 + + , = 0.33161 0.66321 3.76471 + 18.28571 + 8 8 + 8 16 + , = 0.16580 0.33161 1.88235 9.14286 4 4 - + , ζ = 0.00061 + 0.00076 +0.00122 + 0.00152 + 0.00243 + 0.00305 + 0.00488 + 0.00609 +0.00625 + 0.01219 + 0.125 0.02439 + 0.025 + 0.03125 + 0.05 +0.0625 + 0.1 + 0.125 0.2 + 0.25 + 0.5 + 0.5 + – , η = 0.00518 +0.01036 0.05882 + 0.28571 2 + 2 + 0.5 + . Therefore = 2824295299201. IHJPAS. 36 (3) 2023 415 4. Conclusion From our results we get that the Artin indicator equal to the order of the group after we compute the rational valued characters of the rational representations written as a linear combination of the induced characters for the groups . References 1. 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. 2. Saad, O. B. Investigating Particular Representations for Matrix Lie Groups SO(3) and SL(2,₵). Iraqi journal of Science, 2019, 60, 4, 856-858, DOI: 10.24996/ijs.2019.60.4.19. 3. Taghreed, H.; Khawla A.; Niran S. J. “on the Representations of M-Groups”. Baghdad Science journal, 2016, 13, 2, 394-401, DOI:10.21123/bsj.2016.13.2.0394. 4. Sherouk, A. K.; Niran, S. J. “Calculation for the Groups SL(2,U), U = 31 and 37”. 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