440 This work is licensed under a Creative Commons Attribution 4.0 International License IHJPAS. 37 (2) 2024 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq PISSN: 1609-4042, EISSN: 2521-3407 Ghofran Awad Khalaf *1 , Niran Sabah Jasim2 , Zeinab Saeidian3 and Azza I.M.S. Abu-Shams4 1 Ministry of Education, Directorate General of Education in Diyala, Diyala, Iraq. 2 Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham) University of Baghdad, Baghdad, Iraq. 3 University of Kashan, Kashan, Iran. 4 Mathematics of Department, College of Science, Philadelphia University, Amman, Jordan *Corresponding Author. 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, 172) 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. Keywords: Induced characters table, Artin indicator, rational character table of special linear group 𝒮ℒ(n, F). 1. Introduction The representation of the group study by [1], the group of all matrices of determinant 1 is 𝒮ℒ(n,F), [2,3]. Authors in [4,5] study the character table of rational representations for the group 𝒮ℒ(2, 𝑝), and find the periodical split for the groups PSL(2,31) and PSL(2,37) , SL(2,U), U = 31 and 37, we apply the same idea in [6-13] to compute the character table of rational representations for the group 𝒮ℒ(2, 172). Also we apply the same idea in [4,7] to compute the Artin indicator for this group. We count all cyclic subgroup, Artin indicator, the rational valued characters of the rational representations and the induced characters for the group 𝒮ℒ(2, 172) in this work. 2. The Fundamentals Theorem 2.1: [14-16]  𝒮ℒ(2, 𝓆𝓃) = 𝓆𝓃 ( 𝓆2𝓃 – 1). Received: 20 September 2022 Accepted: 28 February 2023 Published: 20 April 2024 Result for the group 𝓢𝓛(2,172) doi.org/10.30526/37.2.3015 https://creativecommons.org/licenses/by/4.0/ https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042 https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407 https://orcid.org/0009-0005-1045-2777 mailto:ghofran.awad1203a@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0001-5340-3020 mailto:niraan.s.j@ihcoedu.uobaghdad.edu.iq https://orcid.org/0000-0002-6279-7217 mailto:saeidian@kashanu.ac.ir https://orcid.org/0009-0003-1741-1447 mailto:aabushams@philadelphia.edu.jo IHJPAS. 37 (2) 2024 441 Definition 2.4: [17-21] Let H be a cyclic subgroup of a group G, and  be a class function of H. Then m GG i i 1H C (g) (g) (x ) C (g)      Definition 2.5: [22-25] The character induced from the principal character of a cyclic subgroups of G is called Artin character. Definition 2.6: [26-30] 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 are Artin characters, is called the Artin exponent of G and denoted by A(G). 3. The Results Authors in [4-7] study the character table of rational representations for the group 𝒮ℒ(2, 𝑝) we apply that idea and compute the character table of rational representations for the group 𝒮ℒ(2, 172). Also we apply the same idea in [4,9] to compute the Artin indicator for this group. The character table of rational representations for the group 𝒮ℒ(2, 172) is Table 1. The character table of rational representations for the group 𝒮ℒ(2, 172) Cg 1 z c = d zc = zd a a2 a3 a4 a6 a8 a9 a12 a16 |Cg | 1 1 41760 41760 83810 83810 83810 83810 83810 83810 83810 83810 83810 |CG(g)| 24137280 24137280 578 578 288 288 288 288 288 288 288 288 288 1G 1 1 1 1 1 1 1 1 1 1 1 1 1 ψ 289 289 0 0 1 1 1 1 1 1 1 1 1 χ1 27840 -27840 96 -96 0 0 0 0 0 0 0 0 0 χ2 6960 6960 24 24 0 0 0 0 0 0 0 0 0 χ3 9280 -9280 32 -32 0 0 0 0 0 0 256 0 512 χ4 3480 3480 12 12 0 0 0 0 0 0 0 144 - 72 χ6 2320 2320 8 8 0 0 0 0 0 64 -32 0 - 64 χ8 1740 1740 6 6 0 0 0 0 36 -48 0 - 72 - 192 χ9 4640 -4640 16 -16 0 0 64 0 -64 0 0 - 128 - 256 χ12 1160 1160 4 4 0 0 0 16 0 -32 -32 - 32 - 32 χ16 870 870 3 3 0 0 9 0 9 -18 -18 - 18 18 χ18 1160 1160 4 4 0 0 -8 0 -16 -32 -32 32 32 χ24 580 580 2 2 0 4 0 -4 -8 -8 8 8 8 χ32 870 870 3 3 0 0 -9 -18 -18 -18 18 18 18 χ36 580 580 2 2 0 0 -4 -8 8 8 8 8 8 χ48 290 290 1 1 1 -1 -2 -2 2 2 2 2 2 χ72 290 290 1 1 0 -2 2 2 2 2 2 2 2 χ96 290 290 1 1 0 -2 2 2 2 2 2 2 2 θ1 32256 -32256 -112 112 0 0 0 0 0 0 0 0 0 θ2 16128 16128 -56 -56 0 0 0 0 0 0 0 0 0 θ5 8064 -8064 -28 28 0 0 0 0 0 0 0 0 0 θ10 4032 4032 -14 -14 0 0 0 0 0 0 0 0 0 θ29 1152 -1152 -4 4 0 0 0 0 0 0 0 0 0 θ52 576 576 -2 -2 0 0 0 0 0 0 0 0 0 ξ 290 290 1 1 -2 2 -2 2 2 2 -2 2 2 η 576 -576 -1 1 0 0 0 0 0 0 0 0 0 IHJPAS. 37 (2) 2024 442 Complete Table 1. The character table of rational representations for the group 𝒮ℒ(2, 172) This group has 26 cyclic subgroups generated by the conjugacy classes of the group. The induced character table for this group is: Table 2. The induced character table for the group 𝒮ℒ(2, 172) Complete Table 2. The induced character table for the group 𝒮ℒ(2, 172) Cg b b 2 b 3 b 6 b 29 58b 87b |Cg | 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 6103437500 |CG(g)| 78126 78126 78126 78126 78126 78126 78126 1G 1 1 1 1 1 1 1 ψ -1 -1 -1 -1 -1 -1 -1 χ1 0 0 0 0 0 0 0 χ2 0 0 0 0 0 0 0 χ3 0 0 0 0 0 0 0 χ4 0 0 0 0 0 0 0 1θ -2 2 4 -4 56 -56 -112 2θ 1 0 -2 0 -28 0 56 3θ 2 2 0 0 -56 56 0 4θ -1 0 0 0 28 0 0 5θ 2 -2 -4 4 0 0 224 6θ -1 0 2 0 0 0 -112 θ7 -2 2 0 0 112 -112 -224 θ8 1 0 0 0 0 0 112 θ 2 -2 -4 4 -56 56 112 θ -1 0 2 -7 28 56 -56 θ -2 2 0 -14 56 -56 -56 θ 1 0 -7 7 -28 -28 -28 θ -2 2 4 -4 -4 -4 -4 θ 1 2 -2 -2 -2 -2 -2 ξ 0 0 0 0 0 0 0 η 4 -4 4 -4 4 -4 4 Cg 1 z c = d zc = zd a a2 a3 a4 a6 a8 a9 a12 a16 |Cg | 1 1 41760 41760 83810 83810 83810 83810 83810 83810 83810 83810 83810 |CG(g)| 24137280 24137280 578 578 288 288 288 288 288 288 288 288 288 Φ1 24137280 0 0 0 0 0 0 0 0 0 0 0 0 Φ2 12068640 12068640 0 0 0 0 0 0 0 0 0 0 0 Φ3 83520 0 2 0 0 0 0 0 0 0 0 0 0 Φ4 83520 41760 2 2 0 0 0 0 0 0 0 0 0 Φ5 83810 167620 0 0 2 0 0 0 0 0 0 0 0 Φ6 41760 0 0 0 0 4 0 0 0 0 0 0 0 Φ7 27840 55680 0 0 0 0 6 0 0 0 0 0 0 Φ8 20880 0 0 0 0 0 0 8 0 0 0 0 0 Φ9 13920 0 0 0 0 0 0 0 12 0 0 0 0 Φ10 10440 0 0 0 0 0 0 0 0 16 0 0 0 Φ11 9280 18560 0 0 0 0 0 0 0 0 18 0 0 Φ12 6960 0 0 0 0 0 0 0 0 0 0 24 0 Φ13 5220 0 0 0 0 0 0 0 0 0 0 0 32 Φ14 4640 0 0 0 0 0 0 0 0 0 0 0 0 Φ15 3480 0 0 0 0 0 0 0 0 0 0 0 0 Φ16 2610 0 0 0 0 0 0 0 0 0 0 0 0 Φ17 2320 0 0 0 0 0 0 0 0 0 0 0 0 Φ18 1740 0 0 0 0 0 0 0 0 0 0 0 0 Φ19 1160 0 0 0 0 0 0 0 0 0 0 0 0 Φ20 870 0 0 0 0 0 0 0 0 0 0 0 0 Φ21 83232 166464 0 0 0 0 0 0 0 0 0 0 0 Φ22 41760 0 0 0 0 0 0 0 0 0 0 0 0 Φ23 16704 33408 0 0 0 0 0 0 0 0 0 0 0 Φ24 8352 0 0 0 0 0 0 0 0 0 0 0 0 Φ25 2880 5760 0 0 0 0 0 0 0 0 0 0 0 Φ26 1440 0 0 0 0 0 0 0 0 0 0 0 0 IHJPAS. 37 (2) 2024 443 Hence, the rational valued characters in the first tables is written as a linear combination of induced characters in the second table 1= 0.008621Φ26 + 0.01724Φ25 + 0.05 Φ24 − 0.1 Φ23 − 0.5 Φ22 −.5 Φ21+ 0.00521 Φ20+ 0.00694 Φ19+ 0.1042 Φ18+ 0.01389 Φ17+ 0.01563 Φ16 + 0.02083 Φ15+ 0.2778 Φ14 + 0.03125 Φ13 + 0.04167 Φ12 + 0.5556 Φ11 + 0,0625 Φ10 + 0.08333 Φ9 + 0.125 Φ8 + 0.16667 Φ7 + 0.25 Φ6 + 0.5 Φ5 + 0.5 Φ4 − 0.01671 Φ2 − 0.00701 Φ1, Ψ = − 0.008621Φ26 − 0.01724Φ25 − 0.05 Φ24+ 0.1 Φ23+ 0.5 Φ22 − 0.5 Φ21+ 0.00521 Φ20+ 0.00694 Φ19+ 0.1042 Φ18+ 0.01389 Φ17+ 0.01563 Φ16+ 0.02083 Φ15+ 0.2778 Φ14 + 0.03125 Φ13 + 0.04167 Φ12 + 0.5556 Φ11 + 0,0625 Φ10 + 0.08333 Φ9 + 0.125 Φ8 + 0.16667 Φ7 + 0.25 Φ6 + 0.5 Φ5 + 0.5 Φ4 − 0.01669 Φ2 + 0.00701 Φ1, χ1 = −24 Φ20 + 48Φ18 − 8 Φ15 + 24 Φ4 + 96 Φ3 + 0.16378Φ2 −0.16753Φ1, χ2 = − 6 Φ20 − 8 Φ19 − 6 Φ18 − 4.5 Φ16 + 12 Φ15 + 12 Φ4 − 0.04095 Φ2 − 0.00004 Φ1, χ3 = 5.33333 Φ20 + 7.11111 Φ19 + 10.66667 Φ18 − 7.11111 Φ17 − 8 Φ16 − 7.11111 Φ14 + 13 Φ13 + 14.22222 Φ11 − 16 Φ4 + 32 Φ3 + 0.03272Φ2 − 0.06129 Φ1 , χ4 = 1.5 Φ20 + 2 Φ19 − 3 Φ18 − 4 Φ17 − 4.5 Φ16 − 3 Φ15 − 2.25 Φ13 + 6 Φ12 + 6 Φ4 + 32 Φ3 − 0.02047 Φ2 − 0.02049 Φ1 , Cg a18 a24 a32 a36 a48 a72 a96 b b2 b5 b10 b29 b58 |Cg | 83810 83810 83810 83810 83810 83810 83810 82943 82943 82943 82943 82943 82943 |CG(g)| 288 288 288 288 288 288 288 290 290 290 290 290 290 Φ1 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ2 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ3 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ4 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ5 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ6 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ7 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ8 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ9 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ10 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ11 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ12 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ13 0 0 0 0 0 0 0 0 0 0 0 0 0 Φ14 36 0 0 0 0 0 0 0 0 0 0 0 0 Φ15 0 48 0 0 0 0 0 0 0 0 0 0 0 Φ16 0 0 64 0 0 0 0 0 0 0 0 0 0 Φ17 0 0 0 72 0 0 0 0 0 0 0 0 0 Φ18 0 0 0 0 96 0 0 0 0 0 0 0 0 Φ19 0 0 0 0 0 144 0 0 0 0 0 0 0 Φ20 0 0 0 0 0 0 192 0 0 0 0 0 0 Φ21 0 0 0 0 0 0 0 2 0 0 0 0 0 Φ22 0 0 0 0 0 0 0 0 2 0 0 0 0 Φ23 0 0 0 0 0 0 0 0 0 10 0 0 0 Φ24 0 0 0 0 0 0 0 0 0 0 20 0 0 Φ25 0 0 0 0 0 0 0 0 0 0 0 58 0 Φ26 0 0 0 0 0 0 0 0 0 0 0 0 116 IHJPAS. 37 (2) 2024 444 χ6 = 0.66667 Φ20 + 0.88889 Φ19 + 1.33333 Φ18 + 1.7778 Φ17 − 2 Φ16 − 2.66667 Φ15 − 1.77778 Φ14 − 2 Φ13 − 1.77778 Φ11 + 4 Φ10 + 4 Φ4 + 0.01092 Φ2 − 0.01452 Φ1 , χ8 = Φ20 + 1.33333 Φ19 +2 Φ18 + 2.66667 Φ17 + 3 Φ16 − 4 Φ15 − 5.33333 Φ14 − 6 Φ13 − 3 Φ12 + 3 Φ9 + 3 Φ4 − 0.01024 Φ2 − 0.0078 Φ1 , χ9 = 1.33333 Φ20 + 1.77778 Φ19 +2.66667 Φ18 + 3.55556 Φ17 + 4 Φ16 − 5.33333 Φ15 − 7.11111 Φ14 − 8 Φ13 − 5.33333 Φ12 − 5.33333 Φ9 + 10.66667 Φ7 − 8 Φ4 + 16 Φ3 − 0.02192 Φ2 − 0.03395 Φ1 , χ12 = 0.6667 Φ20 + 0.22222 Φ19 +0.33333 Φ18 + 0.4444 Φ17 + 0.5 Φ16 + 0.66667 Φ15 + 0.88889 Φ14 − Φ13 − 1.33333 Φ12 − 1.77778 Φ11 − 2 Φ10 + 2 Φ8 + 2 Φ4 − 0.00409 Φ2 − 0.00686 Φ1 , χ16 = 0.09375 Φ20 + 0.125 Φ19 +0.1875 Φ18 + 0.25 Φ17 + 0.28125 Φ16 + 0.375 Φ15 + 0.5 Φ14 + 0.5625 Φ13 − 0.75 Φ12 − Φ11 − 1.125 Φ10 + 0.75 Φ9 + 1.5 Φ7 + 1.5 Φ4 − 0.0105 Φ2 − 0.00658 Φ1 , χ18 = 0.6667 Φ20 + 0.22222 Φ19 +0.33333 Φ18 + 0.4444 Φ17 + 0.5 Φ16 + 0.66667 Φ15 + 0.88889 Φ14 + Φ13 + 1.33333 Φ12 − 1.77778 Φ11 − 2 Φ10 − 1.33333 Φ9 − 1.33333 Φ7 + 2 Φ4 − 0.00206 Φ2 − 0.00383 Φ1 , χ24 = 0.04167 Φ20 + 0.5556 Φ19 +0.08333 Φ18 + 0.01111 Φ17 + 0.125 Φ16 + 0.66667 Φ15 + 0.22222 Φ14 + 0.25 Φ13 + 0.33333 Φ12 + 0.44444 Φ11 − 0.5 Φ10 − 0.66667 Φ9 − 0.5 Φ8 + Φ6 + Φ4−0.00401Φ2 − 0.00455 Φ1 , χ32 = 0.09375 Φ20 + 0.125 Φ19 +0.1875 Φ18 + 0.25 Φ17 + 0.28125 Φ16 + 0.375 Φ15 + 0.5 Φ14 + 0.5625 Φ13 + 0.75 Φ12 + Φ11 − 1.125 Φ10 − 1.5 Φ9 − 2.25 Φ8 − 1.5 Φ7 + 1.5 Φ4 + 0.0018 Φ2 − 0.0010801 Φ1 , χ36 = 0.04167 Φ20 + 0.5556 Φ19 + 0.08333 Φ18 + 0.011111 Φ17 + 0.125 Φ16 + 0.66667 Φ15 + 0.22222 Φ14 + 0.25 Φ13 + 0.33333 Φ12 + 0.44444 Φ11 + 0.5 Φ10 + 0.66667 Φ9 − Φ8 − 0.66667 Φ7 + Φ4 − 0.00102 Φ2 − 0.00282 Φ1 , χ48 =0.01042 Φ20 + 0.01389 Φ19 + 0.2083 Φ18 + 0.02778 Φ17 + 0.03125 Φ16 + 0.04167 Φ15 + 0.05556 Φ14 + 0.625 Φ13 + 0.08333Φ12 + 0.11111Φ11 + 0.125 Φ10 + 0.16667 Φ9 − 0.25Φ8 − 0.33333Φ7 − 0.25 Φ6 + 0.5 Φ5 + 0.5 Φ4 − 0.00728 Φ2 − 0.00268 Φ1 , χ72 = 0.01042Φ20 + 0.01389 Φ19 + 0.2083 Φ18 + 0.02778 Φ17 + 0.03125 Φ16 + 0.04167 Φ15 + 0.05556 Φ14 + 0.625 Φ13 + 0.08333Φ12 + 0.11111Φ11 + 0.125 Φ10 + 0.16667 Φ9 + 0.25Φ8 − 0.33333Φ7 − 0.5 Φ6 + 0.5 Φ4 − 0.00342 Φ2 − 0.00171 Φ1, χ96 =0.01042Φ20 + 0.01389 Φ19 + 0.2083 Φ18 + 0.02778 Φ17 + 0.03125 Φ16 + 0.04167 Φ15 + 0.05556 Φ14 + 0.625 Φ13 + 0.08333Φ12 + 0.11111Φ11 +0.125 Φ10 + 0.16667 Φ9 + 0.25Φ8 − 0.33333Φ7 − 0.5 Φ6 + 0.5 Φ4 − 0.005553 Φ2 − 0.00171 Φ1, IHJPAS. 37 (2) 2024 445 θ1 = 27.03448 Φ26 − 54.06897 Φ25 + 22.4 Φ24 − 44.8 Φ23 − 56 Φ22 − 56 Φ21 + 56 Φ4 − 112 Φ3 − 0.81904 Φ2 + 0.12698 Φ1, θ2 =13.51724 Φ26 + 27.03448 Φ25 − 39.2 Φ24 + 22.4 Φ23 − 28 Φ21 − 28 Φ4 +0.40952 Φ2 + 0.19058 Φ1, θ5 = − 6.75862 Φ26 + 13.51724 Φ25 − 19.6 Φ24 + 14 Φ22 − 14 Φ21 + 14 Φ4 −28 Φ3 + 0.13754 Φ2 + 0.078404 Φ1, θ10 = − 3.37931 Φ26 − 6.75862 Φ25 + 9.8 Φ24 − 19.6 Φ23 + 7 Φ21 − 7 Φ4 − 0.01451 Φ2 − 0.01143 Φ1, θ29 = − 0.13793 Φ26 − 0.27586 Φ25 − 0.8 Φ24 − 1.6 Φ23 + 2 Φ22 − 2 Φ21 + 2 Φ4 −4 Φ3 + 0.02513 Φ2 + 0.01183 Φ1, θ52 = − 0.06897 Φ26 − 0.13793 Φ25 − 0.4 Φ24 − 0.8 Φ23 + 2 Φ22 + Φ21 − Φ4 − 0.00801 Φ2 − 0.00271 Φ1, ζ = 0.01042 Φ20 + 0.01389 Φ19 + 0.2083 Φ18 + 0.02778 Φ17 + 0.03125 Φ16 +0.04167 Φ15 + 0.05556 Φ14 + 0.625 Φ13 + 0.08333Φ12 − 0.11111Φ11 +0.125 Φ10 + 0.16667 Φ9 + 0.25Φ8 − 0.33333 Φ7 + 0.5 Φ6 − Φ5 + 0.5 Φ4 + 0.01369 Φ2 + 0.00089 Φ1, η = −0.03448 Φ26 + 0.06897 Φ25 − 0.2 Φ24 + 0.4 Φ23 − 2 Φ22 + 2 Φ21 + 0.5 Φ4 − Φ3 − 0.0305 Φ2 − 0.00181Φ1. 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