Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
Gikunju David Muriuki^{1}, Nyaga Lewis Namu^{1}, Rimberia Jane Kagwiria^{2}
^{1}Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya
^{2}Department of Pure and Applied Sciences, Kenyatta University, Nairobi, Kenya
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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. 14.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 OrbitStabilizer Theorem [10]
Let be a group acting on a finite set X and x ∈ X. Then
Theorem 2.2 CauchyFrobenius 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