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

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

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.

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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 , 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.

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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

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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 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])

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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 , together with the identity elements. On the other hand if g is elliptic or hyperbolic of order 2, then its centralizer is the 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 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])

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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

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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. of cycles 0 푝푓−1(푞 − 1) 푞+1 (푞−2) 푞 푞 − 1 2 0 푑 푑

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2.3 Representations of G on the cosets of 푯 = 푷풒

To obtain the disjoint cycle structures of 푔 we use Theorem 1.1.9 to get 휋(푔). We compute 훼푙 using Lemma

1.1.10(b). To find 휋(푔) we first need to obtain 퐶푔 and 퐻 ∩ 퐶푔 . Since 퐶푔 is the same as in Section 2.1, so we only need to determine 퐻 ∩ 퐶푔 .

If 푔 ∈ 휏0 and 푔 ∈ 휏2,

퐻 ∩ 퐶푔 = 0.

If 푔 ∈ 휏1 then;

퐻 ∩ 퐶푔 = 푞 − 1,

The values of 흅 품 are displayed in Table 2.3.1 below;

Table 2.3.1: No. of fixed points of elements of G on the cosets of 풑풒

Case 퐶푔 퐶푔 ∩ 퐻 휋 푔

I 푔 ∈ 휏0 푞(푞 − 1) 0 0

2 II 푔 ∈ 휏1 푞 − 1 푞 − 1 푞 − 1

III 푔 ∈ 휏2 푞(푞 + 1) 0 0

We use similar approach to find the disjoint cycle structures of elements of G on the cosets of 푃푞 as section

2.1.

Table 2.3.2: Disjoint Cycle structures of elements of G on the cosets of 풑풒

휏1 휏0 휏2

Cycle length of 푔′ 1 p 1 d 1 d

No. of cycles 푞(푞 − 1) (푞2−1) (푞2 − 1) 푞 − 1 0 0 푝 푑 푑

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3.0 THE CYCLE INDEX FORMULAS FOR 푮 = 푷푮푳(ퟐ, 풒)ACTING ON THE COSETS OF ITS

SUBGROUPS

After computing the disjoint cycle structures of elements 퐺 acting on the cosets of its subgroups, then we can now use them to find the cycle index formula for these representations.

In this section we shall determine some general formulas for finding the cycle indices for the representation of

퐺 on the cosets of its subgroups.

3.1 Cycle index of 푮 acting on the cosets of 푯 = 푪풒−ퟏ

Theorem

The cycle index of 퐺 on the cosets of 퐻 is given by;

퐺 1 푓−1 푍 퐺 = 푡 퐻 + 푞2 − 1 푡푝 푞+1 퐺 1 푝

푞(푞 + 1) 푞−1 푞+2 푞(푞 − 1) 푞 푞+1 + 휑 푑 푡2푡 푑 + 휑 푑 푡 푑 . 2 1 푑 2 푑 1≠푑 푞−1 1≠푑 푞+1

Proof

퐺 퐻 2 The identity contributes 푡1 to the sum of the monomials. All the 푞 − 1 parabolics lie in the same conjugacy class, hence they have the same monomials. So from Table 3.1.2 the contributions by elements of

2 푝푓−1 푞+1 휏1is 푞 − 1 푡푝 . Each 푔 ∈ 휏2 is contained in a unique cyclic subgroup 퐶푞−1 and there are in total

푞(푞+1) conjugates of 퐶 . Hence by Theorem 4.1.1 and the results in Table 3.1.2 the contributions by 2 푞−1

푞−1 푞+2 푞(푞+1) 2 푑 elements of 휏 is 휑 푑 푡 푡 . Finally each 푔 ∈ 휏 is contained in unique cyclic subgroup 2 2 1≠푑 푞−1 1 푑 0

푞(푞−1) 퐶 and there are in total conjugates of 퐶 so they contribute 푞+1 2 푞+1

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푞 푞+1 푞 푞−1 휑 푑 푡 푑 to the same of the monomial. Adding all the contributions and dividing by the 2 1≠푑 푞+1 푑 order of G we get the desired results.∎

3.2 Cycle index of 푮 acting on the cosets of 푯 = 푪풒+ퟏ

Theorem

The cycle index of 푃퐺퐿(2, 푞) acting on the cosets of 퐻 is given by;

퐺 1 푓−1 푍 퐺 = 푡 퐻 + 푞2 − 1 푡푝 푞−1 퐺 1 푝

푞(푞 + 1) 푞 푞−1 푞(푞 − 1) 푞+1 푞−2 + 휑 푑 푡 푑 + 휑 푑 푡2푡 푑 . 2 푑 2 1 푑 1≠푑 푞−1 1≠푑 푞+1

Proof

Using the results in Table 2.2.2 and argument similar to those in Theorem 3.1 the result is immediate. ∎

3.3 Cycle index of 푮 acting on the cosets of 푯 = 푷풒

From the results in Table 2.3.2, we have the following theorem;

Theorem

The cycle index of 퐺 on the cosets of 퐻 is given by;

퐺 1 푓−1 푍 퐺 = 푡 퐻 + 푞2 − 1 푡푞−1푡푝 푞−1 퐺 1 1 푝

1 1 푞(푞 + 1) (푞2−1) 푞(푞 − 1) (푞2−1) + 휑 푑 푡푑 + 휑 푑 푡푑 2 푑 2 푑 1≠푑 푞−1 1≠푑 푞+1

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REFERENCES

Dickson, L. E. (1901). Linear groups with an exposition of the Galois field theory. Teubner, Dover, Newyork

Hardy G. H. and Wright E. M. (1938). An introduction to the theory of numbers. Oxford University Press.

Huppert B. (1967). Endlichegruppen I. Die Grundlehren der Mathematischen Wissenschaften, Band 134. Springer, Berlin.

Kamuti, I. N. (1992). Combinatorial formulas, invariants and structures associated with primitive permutation representations of PSL (2, q) and PGL (2, q), Ph. D. Thesis, Southampton University, U.K.

Redfied, J. H. (1927). The theory of group- reduced distributions. Journal of American Mathematics Society. 49 : 433-455.

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