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Cycle Indices of 푷푮푳 ퟐ, 풒 Acting on the Cosets of Its Subgroups

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Cycle Indices of 푷푮푳 ퟐ, 풒 Acting on the Cosets of Its Subgroups International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 CYCLE INDICES OF 푷푮푳 ퟐ, 풒 ACTING ON THE COSETS OF ITS SUBGROUPS S.K. Rotich Department of Mathematics and Statistics Machakos University P.O. Box 136, Machakos-Kenya Corresponding author E-mail: [email protected] ABSTRACT In this paper, we mainly investigate the action of 푃퐺퐿 2, 푞 on the cosets some of its subgroups namely; 퐶푞−1, 퐶푞+1and 푃푞 . Corresponding to each action the disjoint cycle structures and cycle index formulas is determined. Key words: Disjoint cycle structures, Cycle index formula. 1 Introduction Projective General Linear Group The Projective General Linear Group 푃퐺퐿 2, 푞 over a finite field 퐺퐹 푞 , where 푞 = 푝푓, 푝 a prime number and 푓 a natural number, is a group consisting of all linear fractional transformations of the form; 푎푥 + 푏 푥 → , 푐푥 + 푑 where푥 ∈ 푃퐺 1, 푞 = 퐺퐹 푞 ∪ ∞ , the projective line, 푎, 푏, 푐, 푑 ∈ 퐺퐹 푞 and 푎푑 − 푏푐 ≠ 0. 26 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 It is the factor group of the general linear group by its centre and it is of order 푞 푞2 − 1 . (Dickson [1]) 푃퐺퐿 2, 푞 is the union of three conjugacy classes of subgroups each of which intersects with each other at the identity subgroup, and these are; a) Commutative subgroups of order q 푃퐺퐿(2, 푞)has commutative subgroups of order 푞. We denote it by 푃푞 . Non- identity transformations in 푃푞 have only∞ as a fixed point in 푃퐺(1, 푞). It is of the form; 1 푏 푃 = 푏 ∈ 퐺퐹 푞 , 푞 0 1 The normalizer of 푃푞 in 푃퐺퐿(2, 푞)has transformations of the form; 푐 푒 푆 = 푤푕푒푟푒 푒 ∈ 퐺퐹 푞 푎푛푑 푐, 푑 ∈ 퐺퐹(푞)∗. 0 푑 Thus , 푁푃퐺퐿 2,푞 푃푞 = 푞 푞 − 1 . Therefore the number of subgroups of 푃퐺퐿 2, 푞 conjugate to 푃푞 is 푞 + 1. These 푞 + 1 subgroups have no transformation in common except the identity. All the conjugate subgroups of order 푞 contain 푞2 − 1 distinct non identity transformations of order 푝. 27 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 b) Cyclic subgroups of order 풒 − ퟏ 푃퐺퐿 2, 푞 also contains a cyclic subgroup 퐶푞−1 of order 푞 − 1. Each of its non identity transformations fixes 0 and ∞. It is of the form; 푎 0 퐶 = 푎, 푏 ∈ 퐺퐹(푞)∗ . 푞−1 0 푏 The normalizer of 퐶푞−1 is generated by the transformations 푇 and W 푕 0 Where; 푇 = , 0 1 and h is a primitive element in 퐺퐹 푞 and 0 1 푊 = . 1 0 Thus , 푁푃퐺퐿 2,푞 (퐶푞−1) = 2(푞 − 1). 1 Therefore the number of subgroups of 푃퐺퐿 2, 푞 conjugate to 퐶 is 푞(푞 + 1). These conjugate cyclic 푞−1 2 1 subgroups have no transformation in common except the identity and thus contain 푞(푞 + 1)(푞 − 2) non 2 identity transformations 28 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 c) Cyclic subgroups of order 풒 + ퟏ Another subgroup of 푃퐺퐿 2, 푞 is the cyclic subgroup퐶푞+1 of order 푞 + 1. This consists of elliptic transformations. All the non identity elements of the cyclic subgroup of order 푞 + 1fixes no element. The normalizer of this subgroup is a dihedral subgroup of order 2(푞 + 1). 1 Therefore the number of subgroups of 푃퐺퐿 2, 푞 conjugate to this cyclic subgroup of order 푞 + 1is 푞(푞 − 1). 2 1 1 Since the 푞(푞 − 1) conjugate subgroups have only identity in common so, they contain 푞2(푞 − 1) non 2 2 identity transformations.(Huppert [2]) 1.1 Theorem Let G be a finite transitive permutation group acting on the right cosets of its subgroup H. if 푔휖퐺 and 퐺: 퐻 = 푛 then, 휋 푔 퐶푔 ∩ 퐻 = . 푛 퐶푔 (Kamuti [4]) 1.2 Lemma Let g be a permutation with cycle type ∝1,∝2 ,∝3, … … … . , ∝푛 then, 푙 푙 a) The number 휋 푔 of 1-cycle in 푔 is 푖 푙 푖훼푖 l 1 i b) i ()()gi , where is the mobius function. i il (Hardy and Wright [3]) 29 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 1.3 Lemma If g is elliptic or hyperbolic of order greater than 2 or if g is parabolic, then the centralizer in G consists of all elliptic (respectively, hyperbolic, parabolic) elements with the same points set, together with the identity elements. On the other hand if g is elliptic or hyperbolic of order 2, then its centralizer is the dihedral group of order 2 푞 + 1 or 2 푞 − 1 respectively. (Dickson [1]) 1.4 Theorem a) Let ℘ be the following set of subgroups of 퐺 = 푃퐺퐿 2, 푞 ; 푔 푔 푔 ℘ = {푃푞 , 퐶푞−1, 퐶푞+1 푔 ∈ 퐺}. Then each non-identity elements of G is contained in exactly one group in ℘. (Thus the set ℘ form a partition of G.) b) Let 휋(푔) be the number of fixed points of 푔 ∈ 퐺 on the 푃퐺 1, 푞 . If we define 휏푖 = {푔 푔 ∈ 퐺, 휋 푔 = 푖}; then 푔 푔 푔 휏0 =∪ (푐푞+1 − 퐼) , 휏1 =∪ (푃푞 − 퐼) , 휏2 =∪ (푐푞−1 − 퐼) .(Kamuti [4]) 1.5 Theorem The cycle index of the regular representation of a cyclic group 퐶푛 is given by; 1 푛 푍 퐶 = 휑 푑 푡 푑 , 푛 푛 푑 푑 푛 where휑 is the Euler’s Phi function and 푡1,푡2, … … . , 푡푛 are distinct (commuting) indeterminates. (Redfield [5]) 30 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 2.0 PERMUTATION REPRESENTATIONS OF 푮 = 푷푮푳 ퟐ, 풒 2.1 Representation of G on the cosets of 푯 = 푪풒−ퟏ G contains a cyclic subgroup H of order 푞 − 1whose every non-identity elements fixes two elements. If g is an element in G, we may want to find the disjoint cycle structures of the permutation 푔′ induced by 푔 on the cosets of H.Our computation will be carried out by each time taking an element 푔of order d in G from 휏1,휏2 and 휏0 respectively. To find the disjoint cycle structuresof 푔 ∈ 퐺we use Lemma 1.2(b) and thus we need to determine 퐶푔 , 퐶푔 ∩ 퐻 and 휋푔 using Theorem 1.1 We easily obtain 퐶푔 using Lemma 1.3, but we need to distinguish between 푑=2 and 푑 > 2. So if 푑 > 2 then we have; 퐺 퐶푔 = , 퐶(푔) where 퐶(푔) is the order of centralizer of g. The values of 휋 푔 are displayed in Table 2.1.1 below Table 2.1.1: No. of fixed points of elements of G acting on the cosets of 푪풒−ퟏ Case 퐶푔 퐶푔 ∩ 퐻 휋 푔 I 푔 ∈ 휏0푑 > 2 푞(푞 − 1) 0 0 푑 = 2 푞(푞 − 1) 0 0 2 2 II 푔 ∈ 휏1 푑 > 2 푞 − 1 0 0 푑 = 2 푞2 − 1 0 0 III 푔 ∈ 휏2 푑 > 2 푞(푞 + 1) 2 2 푑 = 2 푞(푞 + 1) 1 2 2 After obtaining 휋 푔 , we proceed to calculate in details the disjoint cycles structures of elements푔′ in this representation using Lemma 1.2(b). 31 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 Table 2.1.2: Disjoint cycle structures of elements of G on the cosets of 푪풒−ퟏ 휏1 휏0 휏2 Cycle length of 1 p 1 d 1 d 푔′ 푓−1 푞(푞+1) No. of cycles 0 푝 (푞 + 1) 0 푞 − 1 (푞 + 2) 푑 2 푑 2.2 Representation of G on the cosets of 푯 = 푪풒+ퟏ To compute the disjoint cycle structures of 푔′ we also need to determine 휋(푔) by using Theorem 1.1 and find 푔 푔 푔 훼푙 by applying Lemma 1.2(b). Before we obtain 휋(푔) we first need to determine 퐶 and 퐶 ∩ 퐻 . Since 퐶 푔 is the same as in Section 2.1, so we only need to find 퐶 ∩ 퐻 . If 푔휖휏1 and 휏2then, 퐶푔 ∩ 퐻 = 0. If 푔휖휏0 then, 1 푖푓 푑 = 2 퐶푔 ∩ 퐻 = . 2 푖푓 푑 ≠ 2 This is because if푑 = 2 휑 2 = 1. So it has a single element of order 2. Each subgroup of G conjugate to퐻 has one element of order 2. If 푔휖휏0 and 푑 ≠ 2we have; 퐶푔 ∩ 퐻 = 2 32 International Journal of Scientific Research and Innovative Technology ISSN: 2313-3759 Vol. 5 No. 4; April 2018 The values of 휋 푔 are displayed in Table 2.2.1 below Table 2.2.1: No. of fixed points of elements of G acting on the cosets of 푪풒+ퟏ Case 퐶푔 퐶푔 ∩ 퐻 휋 푔 I 푔 ∈ 휏0푑 > 2 푞(푞 − 1) 2 2 푑 = 2 푞(푞 − 1) 1 2 2 2 II 푔 ∈ 휏1 푑 > 2 푞 − 1 0 0 푑 = 2 푞2 − 1 0 0 III 푔 ∈ 휏2 푑 > 2 푞(푞 + 1) 0 0 푑 = 2 푞(푞 + 1) 0 0 2 After obtaining 휋(푔) we use the same approach as in Section 2.1 above to find the disjoint cycle structures of elements of G on the cosets of 퐶푞+1.Therefore the results is as shown in Table 2.2.2 below. Summary of results Table 2.2.2: Disjoint Cycle structures of elements of G on the cosets of 푪풒+ퟏ 휏1 휏0 휏2 Cycle length of 푔′ 1 p 1 d 1 d No.
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