Pure and Applied Mathematics Journal
Volume 6, Issue 1, February 2017, Pages: 1-4

Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Gikunju David Muriuki1, Nyaga Lewis Namu1, Rimberia Jane Kagwiria2

1Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

2Department of Pure and Applied Sciences, Kenyatta University, Nairobi, Kenya

Email address:

(Gikunju D. M.)

To cite this article:

Gikunju David Muriuki, Nyaga Lewis Namu, Rimberia Jane Kagwiria. Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Pure and Applied Mathematics Journal.Vol. 6, No. 1, 2017, pp. 1-4.doi: 10.11648/j.pamj.20170601.11

Received: December 17, 2016; Accepted: January 3, 2017; Published: February 2, 2017


Abstract: Transitivity and Primitivity of the action of the direct product of the symmetric group on Cartesian product of three sets are investigated in this paper. We prove that this action is both transitive and imprimitive for all . In addition, we establish that the rank associated with the action is a constant  Further; we calculate the subdegrees associated with the action and arrange them according to their increasing magnitude.

Keywords: Direct Product, Symmetric Group, Action, Rank, Subdegrees, Cartesian Product, Suborbit


1. Introduction

Group Action of on  is defined as   . This paper explores the action  on

2. Notation and Preliminary Results

Let be a set, a group  acts on the left ofif for each and  there corresponds a unique element such that , and and, , where  is the identity in.

The action of  on  from the right can be defined in a similar way.

Let act on a set . Then  is partitioned into disjoint equivalence classes called orbits or transitivity classes of the action. For each  the orbit containing x is called the orbit of  and is denoted by Thus

.

The action of a group on the set is said to be transitive if for each pair of points, there exists g G such that; in other words, if the action has only one orbit.

Suppose that acts transitively on X. Then a subset Y of , where |Y| is a factor of |X|, is called a block or set of imprimitivity for the action if for each g G, either gY=Y or gY∩Y= ; in other words gY and Y do not overlap partially. In particular, , X and all 1- element subsets of  are obviously blocks. These are called the trivial blocks. If these are the only blocks, then we say that G acts primitively on . Otherwise, G acts imprimitively.

Theorem 2.1 Orbit-Stabilizer Theorem [10]

Let be a group acting on a finite set X and x X. Then

Theorem 2.2 Cauchy-Frobenius lemma [4]

Let G be finite group acting on a set X. The number of orbits of G is .

Let and be permutation groups. The direct product  acts on Cartesian product  by rule.

3. Main Results

3.1. Transitivity of  Acting on

Theorem 3.1 If n ≥ 2, then =  acts transitively on , where, and.

Proof. Let  act on . It is enough to show that the cardinality of

 is equal to. We determine.

Now fixes

If and only if so that,, . Thus comes from a 1- cycle of.

Hence  is isomorphic to and so

By the use Theorem 2.1

=

=

=

3.2. Primitivity of  on

Theorem 3.2 The action of on is imprimitive for.

Proof. This action is transitive by theorem 3.1 Consider  where

, and  therefore  acts on and . Let be any non trivial subset of  such that  divides by  therefore for . For each element of  there exist with  cycles permutation for  such that moves an element of to an element not in  so that . This argument shows that  is a block for the action and the conclusion follows.

3.3. Ranks and Subdegrees of  on

In this section , and

Theorem 3.3 The rank of  acting on  is .

Proof. Let  act on.

, the identity. By use of Theorem 2.5 to get the number of orbits of  on

Let, then

A permutation in  is of the form (1,1,1 ) since it is the identity. The number of elements in fixed by each is 8 since identity fixes all the elements in.

Hence by Cauchy Frobenius Lemma, the number of orbits of  acting on is

Let

The orbits of  on  are:

a) The Suborbit whose every element contains exactly 3 elements from  -the trivial orbit.

b) Suborbits each of whose every element contains exactly 2 elements from

c) Suborbits each of whose every element contains exactly 1 element from

d) Suborbit whose elements contain no element from

From the above, the rank of  acting on  is and subdegrees are 1,1,...,1.

3.4. Ranks and Subdegrees of  on

In this section , and

Theorem 3.4 The rank of  acting on  is .

Proof. Let act on  and  then

 

By Theorem 2.1,

By applying Cauchy Frobenius Lemma the number of orbits of acting on  are;

The number of elements in fixed by each is given by Table1

Table 1. Permutations in and the number of fixed points.

Type of ordered triple permutations in Number of ordered triple of permutations

1 27

1 9

1 9

1 3

1 9

1 3
  1 3

1 1

Now applying Theorem 2.2, the number of orbits of  acting on  is

 

Let

The  orbits of  on  are:

a) Suborbit whose every element contains exactly 3 elements from

-the trivial orbit.

b) Suborbits each of whose every element contains exactly 2 elements from

c) Suborbits each of whose every element contains exactly 1 element from

d) Suborbit whose elements contain no element from

From the above, the rank of  acting on is  and subdegrees are 1,2,2,2,4,4,4,8

3.5. Ranks and Subdegrees of  on

Theorem 3.5 If , the rank ofacting onis , where

,  and.

Proof. The number orbits of acting onare given as follows;

Let

Table 2. The rank of  acting on.

Suborbit   Number of suborbits

Orbit containing no element from

1

Orbits containing exactly  element from

2

Orbits containing exactly 2 elements from

2

Orbit containing exactly 3 elements from

1

Hence the rank of the acting onis

The orbits of  on  are:

a) Suborbit whose every element contains exactly 3 elements from

 -the trivial orbit.

b) Suborbits each of whose every element contains exactly 2 elements from

 .

 .

 .

c) Suborbits each of whose every element contains exactly 1 element from

.

 .

 .

d) Suborbit whose elements contain no element from

 .

The subdegrees of  are as shown in Table 3 below:

Table 3. Subdegrees of  acting on for n ≥ 2.

Suborbit length 1

number of suborbits 1 3 3 1

4. Conclusions

In this study, some properties of the action of  on  were studied. It can be concluded that;

 acts transitively and imprimitely on  for all.

The rank of on  is  for all.


References

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