Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Gikunju David Muriuki; Nyaga Lewis Namu

doi:doi:10.11648/j.ajam.20241205.17

Research Article |

| Peer-Reviewed

Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Gikunju David Muriuki^*

, Nyaga Lewis Namu

Published in American Journal of Applied Mathematics (Volume 12, Issue 5)

Received: 26 August 2024 Accepted: 25 September 2024 Published: 18 October 2024

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)², (n-1)², (n-1)², (n-1)³. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ₂ and Γ₃ are regular of degree n-1, graphs Γ₄, Γ₅ and Γ₆ of degree (n-1)² and graph Γ₇ is regular of degree (n-1)³. The suborbital graphs Γ_i(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n² while suborbital graph Γ₇ is connected for all n > 2.

Published in	American Journal of Applied Mathematics (Volume 12, Issue 5)
DOI	10.11648/j.ajam.20241205.17
Page(s)	175-182
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Rank, Subdegrees, Suborbit, Suborbital Graphs

1. Introduction

The group action of the direct product of the symmetric group on Cartesian product of three sets i.e.

S_{n} \times S_{n} \times S_{n}

X \times Y \times

is defined as

(g_{1}, g_{2}, g_{3}) (x, y, z) = (g_{1} x, g_{2} y, g_{3} z) \forall g_{1}, g_{2}, g_{3} \in S_{n}

x \in X, y \in Y, z \in Z

[1]

. This paper investigates properties of the suborbital graphs that arise from this group action.

The degree of a vertex in any graph is the number of vertices adjacent to that vertex. A vertex is isolated if it has degree 0

[6]

The length of the shortest cycle in graph is called its girth.

A connected graph is one where every pair of vertices is connected by a path; otherwise, it is disconnected. A maximal connected subgraph of a graph is a connected component of that graph. In this case every vertex and edge of the graph belongs to exactly one connected component of the graph

[2]

If a group G acts transitively on a set X and if Δ is an orbit of the stabilizer of G on

X

, then

Δ^{*} = \{gx| g \in G, x \in g Δ\} .

In this case Δ * is also an orbit of the stabilizer of G and is called the G -suborbit paired with Δ

[7]

. Clearly | Δ |=| Δ * | and Δ ** = Δ. If

Δ^{*} = Δ

, then

Δ

is said to be self-paired

[8]

Theorem 1.1(Sims Theorem) If

Γ^{*} {}_{i}

are the suborbital graphs corresponding to the suborbital

{ο_{i}}^{*}_{}

. If the suborbits

correspond to the suborbital

{ο_{i}}_{}

. Then

Γ_{i}

is undirected if

Δ_{i}

is self-paired and

Γ_{i}

is directed if

Δ_{i}

is not self-paired

[7]

2. Main Results

2.1. Ranks and Subdegrees

Theorem 2.1 The rank of

G = S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

is 8, for all

n \geq 2

Proof. Let

X = {1, 2, . . ., n}, Y = {n + 1, n + 2, . . ., 2 n}

and

Z = \{2 n + 1, 2 n + 2, . . ., 3 n\}

. The number orbits of

G = S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

are given as follows;

Let

B = \{1, n + 1, 2 n + 1\}

then

Table 1. The rank of

S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

Suborbit	Formula	Number of suborbits
Orbit comprising no element from $B$	³C₀	1
Orbits comprising a single element from $B$	³C₁	3
Orbits comprising two elements from $B$	³C₂	3
Orbit comprising three elements from $B$	³C₃	1

Hence the rank of the rank of

G = S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

1 + 3 + 3 + 1 = 8

[3]

The 8 orbits of

G_{(1, n + 1, 2 n + 1)}

X \times Y \times Z

are:

a) Suborbit whose every element comprises

3

elements from

B

∆_{0} = {Orb}_{G (1, n + 1, 2 n + 1)} (1, n + 1, 2 n + 1) = {(1, n + 1, 2 n + 1)}

-the trivial orbit.

b) Suborbits whose every element contains

2

elements from

B

∆_{1} = {Orb}_{G (1, n + 1, 2 n + 1)} (1, n + 1, 2 n + 2) = {(1, n + 1, 2 n + 2), (1, n + 1, 2 n + 3), \dots, (1, n + 1, 3 n)\}, n \geq 2 .

∆_{2} = {Orb}_{G (1, n + 1, 2 n + 1)} (1, n + 2, 2 n + 1) = {(1, n + 2, 2 n + 1), (1, n + 3, 2 n + 1), \dots, (1, 2 n, 2 n + 1)\}, n \geq 2 .

∆_{3} = {Orb}_{G (1, n + 1, 2 n + 1)} (2, n + 1, 2 n + 1) = {(2, n + 1, 2 n + 1), (3, n + 1, 2 n + 1), \dots, (n, n + 1, 2 n + 1)\}, n \geq 2 .

c) Suborbits whose every element contains three elements from

B

∆_{4} = {Orb}_{G (1, n + 1, 2 n + 1)} (1, n + 2, 2 n + 2) = {(1, n + 2, 2 n + 2), (1, n + 3, 2 n + 3), \dots, (1, 2 n, 3 n)\}, n \geq 2 .

∆_{5} = {Orb}_{G (1, n + 1, 2 n + 1)} (2, n + 1, 2 n + 2) = {(2, n + 1, 2 n + 2), (3, n + 1, 2 n + 3), \dots, (n, n + 1, 3 n)\}, n \geq 2 .

∆_{6} = {Orb}_{G (1, n + 1, 2 n + 1)} (2, n + 2, 2 n + 1) = {(2, n + 2, 2 n + 1), (3, n + 2, 2 n + 1), \dots, (n, 2 n, 2 n + 1)\}, n \geq 2 .

d) Suborbit containing no element from

B

∆_{7} = {Orb}_{G (1, n + 1, 2 n + 1)} (2, n + 2, 2 n + 2) = {(2, n + 2, 2 n + 2), (3, n + 2, 2 n + 2), \dots, (n, 2 n, 3 n)\}, n \geq 2 .

Table 2. The subdegrees of S_n×S_n×S_n acting on X×Y×Z.

Suborbit length	1	$n - 1$	${(n - 1)}^{2}$	${(n - 1)}^{3}$
number of suborbits	1	3	3	1

2.2. Suborbital Graphs of S₂×S₂×S₂ Acting on X×Y×Z and Their Properties

Theorem 2.2 All Suborbits of

S_{2} \times S_{2} \times S_{2}

acting on

X \times Y \times Z

are self-paired.

Proof. It was proven that the number of orbits of

G_{(1, 3, 5)}

acting on X×Y×Z is

8

orbits. The suborbital graph corresponding to

is a null graph. The remaining suborbits

Δ_{1}, Δ_{2},..., Δ_{7}

are self- paired. Consider for the example

= {(1, 4, 5)}for

(g_{1}, g_{2}, g_{3}) \in G

⇒

(g_{1}, g_{2}, g_{3}) (1, 4, 5) = (1, 3, 5)

⇒

g_{1} (1) = 1

g_{2} (4) = 3

and

g_{3} (5) = 5

, therefore this group action can be given by

(g_{1}, g_{2}, g_{3}) = (1, (4 3), (5)) .

⇒

(g_{1}, g_{2}, g_{3}) (1, 3, 5) = (1, 4, 5) \in Δ_{2}

⇒

{Δ^{*}}_{2} = Δ_{2}

Therefore, self-paired.

By Theorem 1.1, their corresponding suborbital graphs

Γ_{1}, Γ_{2}, . . ., Γ_{7}

are undirected due to the fact that all suborbits are self-paired.

The suborbital graph corresponding to

is a null graph and the remaining seven suborbital graphs are given as follows;

Assume that

A

and

B

are two distinct points from

X \times Y \times Z

. Then the corresponding graphs are constructed as follows;

The suborbital

ο_{1} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (1, 3, 6) | (g_{1}, g_{2}, g_{3}) \in G\}

Consequently, in graph

Γ_{1}

, the suborbital graph corresponding to

ο_{1}

, there is an edge from

A

B

iff the first two coordinates of

A

are identical to the first two coordinates of

B

and the third coordinate of

A

is not identical to the third coordinate of

B

[5]

Download: Download full-size image

Figure 1. The suborbital graph

Γ_{1}

Properties of the graph

Γ_{1}

1. The graph has no cycle and hence a girth of zero.

2. The graph is disconnected.

3. The graph is a regular of degree

1

The suborbital

ο_{2} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (1, 4, 5) | (g_{1}, g_{2}, g_{3}) \in G\}

Consequently, in graph

Γ_{2}

, the suborbital graph corresponding to

ο_{2}

, there is an edge from

A

B

iff the first and third coordinates of

A

is identical to the first and third coordinates of

B

and the second coordinate of

A

is not identical to the second coordinate of

B

Download: Download full-size image

Figure 2. The suborbital graph

Γ_{2}

Properties of graph

Γ_{2}

1. The graph has no cycle and hence has a girth zero.

2. The graph is disconnected.

3. The graph is regular of degree

1

The suborbital

ο_{3} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (2, 3, 5) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{3}

, the suborbital graph corresponding to

ο_{3}

, there is an edge from

A

B

iff the last two coordinates of

A

are identical to the last two coordinates of

B

and the first coordinate of

A

is not identical to the first coordinate of

B .

Download: Download full-size image

Figure 3. The suborbital graph

Γ_{3}

Properties of graph

Γ_{3}

i. The graph has a girth zero.

ii. The graph is disconnected.

iii. The graph is regular of degree

1

The suborbital

ο_{4} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (1, 4, 6) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{4}

, the suborbital graph corresponding to

ο_{4}

, there is an edge from

A

B

iff the first coordinate of

A

is identical to the first coordinate of

B

and the last two coordinates of

A

are not identical to the last two coordinates of

B

Download: Download full-size image

Figure 4. The suborbital graph

Γ_{4}

Properties of graph suborbital

Γ_{4}

1. The graph has a girth zero.

2. The graph is disconnected.

3. The graph is regular of degree

1

The suborbital

ο_{5} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (2, 3, 6) | (g_{1}, g_{2}, g_{3}) \in G\}

Consequently, in graph

Γ_{5}

, the suborbital graph corresponding to

ο_{5}

, there is an edge from A to

B

iff the second coordinates of

A

are identical to the second coordinate of

B

and the first and third coordinates of

A

are not identical to the first and third coordinates of

B

Download: Download full-size image

Figure 5. The suborbital graph

Γ_{5}

Properties of graph

Γ_{5}

1. The graph has a girth zero.

2. The graph is disconnected.

3. The graph is regular of degree

1

The suborbital

ο_{6} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (2, 4, 5) | (g_{1}, g_{2}, g_{3}) \in G\}

Thus, in graph

Γ_{6}

, the suborbital graph corresponding to

ο_{6}

, there is an edge from

A

B

iff the last coordinate of

A

is identical to the last coordinate of B and the first two coordinates of

A

is not identical to the first two coordinates of

B

The properties of the graph

Γ_{6}

are; the graph has no cycle and hence has a girth zero. It is disconnected, regular of degree 1.

Download: Download full-size image

Figure 6. The suborbital graph

Γ_{6}

The suborbital

ο_{7} = \{(g_{1}, g_{2}, g_{3}) (1, 3, 5) (g_{1}, g_{2}, g_{3}) (1, 4, 6) | (g_{1}, g_{2}, g_{3}) \in G\}

Thus in graph

Γ_{7}

, the suborbital graph corresponding to

ο_{7}

, there is an edge from

A

to B iff there is no coordinate of

A

is identical to any coordinate of

B

[4]

Download: Download full-size image

Figure 7. The suborbital graph

Γ_{7}

Properties of The graph

Γ_{7}

a) The graph has no cycle and hence has a girth zero.

b) It is disconnected.

c) It is regular of degree

1

2.3. Suborbital Graphs of S₃×S₃×S₃ Acting on X×Y×Z and their Properties

As seen in the previous section, the suborbital graph corresponding to

is a null graph and the remaining seven suborbital graphs of S₃×S₃×S₃ Acting on X×Y×Z are given as follows;

Assume that

A

and

B

are two distinct points from

X \times Y \times Z

. Then the graphs are constructed as follows;

The suborbital

ο_{1} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (1, 4, 8) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{1}

, the suborbital graph corresponding to

ο_{1}

, there is an edge from

A

B

iff the first two coordinates of A are identical to the first two coordinates of

B

and the third coordinate of

A

is not identical to the third coordinate of

B

Download: Download full-size image

Figure 8. The suborbital graph

Γ_{1}

Properties of the graph

Γ_{1}

1. The graph is disconnected graph with

9

connected components.

2. It is regular of degree

2

3. It has a girth three.

The suborbital

ο_{2} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (1, 5, 7) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{2}

, the suborbital graph corresponding to

ο_{2}

, there is an edge from

A

B

iff the first and third coordinates of A is identical to the first and third coordinates of B and the second coordinate of

A

is not identical to the second coordinate of

B

Download: Download full-size image

Figure 9. The suborbital graph

Γ_{2}

Properties of the graph

Γ_{2}

1. The graph is disconnected graph with

9

connected components

2. It is regular of degree

2

3. It has a girth three.

The suborbital

ο_{3} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (2, 4, 7) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{3}

, the suborbital graph corresponding to

ο_{3}

, there is an edge from A to B iff the last two coordinates of A are identical to the last two coordinates of B and the first coordinate of A is not identical to the first coordinate of B.

Download: Download full-size image

Figure 10. The suborbital graph

Γ_{3}

Properties of the graph

Γ_{3}

1. The graph is disconnected graph with 9 connected components.

2. The graph is regular of degree 2.

3. The graph has a girth three.

The suborbital

ο_{4} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (1, 5, 8) | (g_{1}, g_{2}, g_{3}) \in G\}

Therefore, in graph

Γ_{4}

, the suborbital graph corresponding to

ο_{4}

, there is an edge from A to B iff the first coordinate of A is identical to the first coordinate of B and the last two coordinates of A are not identical to the last two coordinates of B.

Download: Download full-size image

Figure 11. The suborbital graph

Γ_{4}

Properties of the graph

Γ_{4}

1. The graph is disconnected graph with 9 connected components

2. It is regular of degree 4.

3. It has a girth three.

The suborbital

ο_{5} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (2, 4, 8) | (g_{1}, g_{2}, g_{3}) \in G\}

Thus in

Γ_{5}

, the suborbital graph corresponding to

ο_{5}

, there is an edge from A to B iff the second coordinates of A are identical to the second coordinate of B and the first and third coordinates of A are not identical to the first and third coordinates of B.

Download: Download full-size image

Figure 12. The suborbital graph

Γ_{5}

Properties of the graph

Γ_{5}

1. The graph is disconnected.

2. It is regular of degree 4.

3. It has a girth three.

The suborbital

ο_{6} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (2, 5, 7) | (g_{1}, g_{2}, g_{3}) \in G\}

Thus in

Γ_{6}

, the suborbital graph corresponding to

ο_{6}

, there is an edge from A to B iff the last coordinate of A is identical to the last coordinate of B and the first two coordinates of A is not identical to the first two coordinates of B.

Download: Download full-size image

Figure 13. The suborbital graph

Γ_{6}

corresponding to the suborbit

Δ_{6}

S_{3} \times S_{3} \times S_{3}

acting on

X \times Y \times Z

Properties of the graph

Γ_{6}

1. The graph is disconnected.

2. It is regular of degree 4.

3. It has a girth three.

The suborbital

ο_{7} = \{(g_{1}, g_{2}, g_{3}) (1, 4, 7) (g_{1}, g_{2}, g_{3}) (2, 5, 8) | (g_{1}, g_{2}, g_{3}) \in G\}

Thus in

Γ_{7}

, the suborbital graph corresponding to

ο_{7}

, there is an edge from A to B iff there is no coordinate of

A

is identical to any coordinate of

B .

Download: Download full-size image

Figure 14. The suborbital graph

Γ_{7}

Properties of the graph

Γ_{7}

1. the graph is connected.

2. It is regular of degree.

3. Ithas a girth three.

2.4. Suborbital Graphs of S_n×S_n×S_n Acting on X×Y×Z and Their Properties

Theorem 2.4.1 Suborbits

Δ_{1}, Δ_{2},..., Δ_{7}

S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

are self-paired.

Proof. The proof of

Δ_{0}

is trivial.

Consider

Δ_{1} = {(1, n+1,2n +2),..., (1, n+1,3n)}

\forall n \geq 2

is self-paired for if

(g_{1}, g_{2}, g_{3}) {(1, n+1,2n+2),..., (1, n+1,3n)}

. Consider

g_{3} {2n+2,2n +3,...,3n}

since the rest are constant. Thus

g_{3} {2n+2,2n+3,...,3n} = {1,2,..., n-1}

, then take

g_{3}

to be

(2n+21) (2n+32),... (3nn-1)

and

g_{3} {1,2,..., n-1} = {2n+2,2n+3,...,3n} \in Δ_{1} .

This shows that

Δ_{1}

is self-paired.

Similarly, consider

Δ_{2} = {(1, n+2,2n +1),..., (1,2n,2n+1)} \forall n \geq 2

. If

(g_{1}, g_{2}, g_{3}) {(1, n+2,2n+1),..., (1,2n,2n+1)}

Consider

g_{2} {2n+2,2n +3,...,3n}

since the rest are constant. Thus

g_{2} {2n+2,2n+3,...3n} = {1,2,..., n-1}

, then take

g_{2}

to be

(2n+21) (2n+32),..., (3nn-1)

and

g_{2} {1,2,..., n-1} = {2n+2,2n+3,...,3n} \in Δ_{2}

This show that

Δ_{2}

is also self-paired. With similar arguments

Δ_{3}, Δ_{4}, Δ_{5}

and

Δ_{6}

are self-paired.

Finally, consider

Δ_{7} = {(2, n+2,2n +2),..., (n,2n,3n)} \forall n \geq 2

. If

(g_{1}, g_{2}, g_{3}) {(2, n+2,2n+2),..., (n,2n,3n)}

consider

g_{1} {2,3,..., n}

since for the cases of

g_{2}

and

g_{3}

are already shown that are self-paired. Thus

g_{1} {2,3,...n} = {1,2,..., n-1}

, then take

g_{1}

to be

(21), (32),..., (nn-1)

and

g_{1} {1,2,..., n-1} = {2,3,..., n} \in Δ_{7}

. Showing that

Δ_{7}

is a self-paired.

Theorem 2.4.2 All the suborbitals graphs of G are undirected.

Proof. From Theorem 1.1 and Theorem 2.4.1 the following results are straight forward due to the fact that all suborbits are self-paired.

Corollary 2.4.3 Let

G = S_{n} \times S_{n} \times S_{n}

act on

X \times Y \times Z

then the suborbital graph

Γ_{i} (i = 1, 2, . . ., 6)

is disconnected with the number of connected components equal to

n^{2}

and suborbital graph

Γ_{7}

is connected for all

n > 2

Corollary 2.4.4 Let

G = S_{n} \times S_{n} \times S_{n}

act on

X \times Y \times Z

then the suborbital graph

Γ_{i} (i = 1, 2, . . ., 7)

has a girth of 3 if

n > 2

and a girth is zero if

n=2

Corollary 2.4.5 Let

G = S_{n} \times S_{n} \times S_{n}

act on

X \times Y \times Z

graphs

Γ_{1}, Γ_{2}

and

Γ_{3}

are regular of degree

n-1

, graphs

Γ_{4}, Γ_{5}

and

Γ_{6}

are regular of degree

{(n - 1)}^{2}

and graph

Γ_{7}

is regular of degree

{(n - 1)}^{3}

. This agrees with subdegrees of

S_{n} \times S_{n} \times S_{n}

acting on

X \times Y \times Z

Abbreviations

$∆$	The Suborbit of G on X
$∆^{*}$	$The G - suborbit paired with ∆$
$Γ$	The Suborbital Graph Corresponding to the Suborbit $∆$
$X \times Y \times Z$	$The Cartesian product of sets X, Y and Z$
$S_{n} \times S_{n} \times S_{n}$	External direct product of $S_{n}$

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Cameron, P. J., Gewurz, D. A., and Merola, F. (2008). Product action. Discrete Math. 386–94. https://www.sciencedirect.com/science/article/pii/S0012365X07003810
[2]	Harary, F. (1969). Graph Theory. Addison-Wesley Publishing Company, New York. https://www.sciencedirect.com/science/article/abs/pii/0378873383900266
[3]	Muriuki, G. D., Namu, N. L., & Kagwiria, R. J. (2017). Ranks, sub degrees and suborbital graphs of direct product of the symmetric group acting on the cartesian product of three sets. Pure and Applied Mathematics Journal, 6(1), 1. https://www.researchgate.net/profile/David-Gikunju/publication/316219152_Ranks_Subdegrees_and_Suborbital_Graphs_of_Direct_Product_of_the_Symmetric_Group_Acting_on_the_Cartesian_Product_of_Three_Sets/links/5d1713b2458515c11c009d97/Ranks-Subdegrees-and-Suborbital-Graphs-of-Direct-Product-of-the-Symmetric-Group-Acting-on-the-Cartesian-Product-of-Three-Sets.pdf
[4]	Nyaga, L. N. (2012). Ranks, Subdegrees and Suborbital Graphs of the Symmetric Group S_nActing on Unordered r−Element Subsets. PhD thesis, JKUAT, Juja, Kenya. https://www.ajol.info/index.php/jagst/article/view/112798
[5]	Rimberia, J. K. (2011). Ranks and Subdegrees of the Symmetric Group S_n Acting on Ordered r− Element Subsets. PhD thesis, Kenyatta University, Nairobi, Kenya. https://ir-library.ku.ac.ke/bitstream/handle/123456789/3970/Rimberia%20Jane%20Kagwiria%20.pdf?sequence=3&isAllowed=y
[6]	Rose, J. S. (1978). A Course in Group Theory. Cambridge University Press, Cambridge. https://books.google.co.ke/books?hl=en&lr=&id=j-I7Zpq3GdIC&oi=fnd&pg=PP1&dq=Rose,+J.+S.+(1978).+A+Course+in+Group+Theory.+Cambridge+University+Press,+Cambridge.&ots=bagHM7veAF&sig=e0pmlEQuZZ6X6G0q8G2cIDO7ZKk&redir_esc=y#v=onepage&q&f=false
[7]	Sims, C. C. (1967). Graphs and finite permutation groups. Mathematische Zeitschrift, 95: 76–86. https://link.springer.com/article/10.1007/BF01117534
[8]	Wielandt, H. (1964). Finite Permutation Groups. Academic Press New York. https://www.scirp.org/(S(czeh2tfqw2orz553k1w0r45))/reference/referencespapers.aspx?referenceid=2087842

Cite This Article

Plain Text BibTeX RIS

APA Style

Muriuki, G. D., Namu, N. L. (2024). Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. American Journal of Applied Mathematics, 12(5), 175-182. https://doi.org/10.11648/j.ajam.20241205.17

Copy | Download

ACS Style

Muriuki, G. D.; Namu, N. L. Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Am. J. Appl. Math. 2024, 12(5), 175-182. doi: 10.11648/j.ajam.20241205.17

Copy | Download

AMA Style

Muriuki GD, Namu NL. Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Am J Appl Math. 2024;12(5):175-182. doi: 10.11648/j.ajam.20241205.17

Copy | Download

@article{10.11648/j.ajam.20241205.17,
  author = {Gikunju David Muriuki and Nyaga Lewis Namu},
  title = {Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {5},
  pages = {175-182},
  doi = {10.11648/j.ajam.20241205.17},
  url = {https://doi.org/10.11648/j.ajam.20241205.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241205.17},
  abstract = {The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.
},
 year = {2024}
}

Copy | Download

TY - JOUR
T1 - Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

AU - Gikunju David Muriuki
AU - Nyaga Lewis Namu
Y1 - 2024/10/18
PY - 2024
N1 - https://doi.org/10.11648/j.ajam.20241205.17
DO - 10.11648/j.ajam.20241205.17
T2 - American Journal of Applied Mathematics
JF - American Journal of Applied Mathematics
JO - American Journal of Applied Mathematics
SP - 175
EP - 182
PB - Science Publishing Group
SN - 2330-006X
UR - https://doi.org/10.11648/j.ajam.20241205.17
AB - The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.

VL - 12
IS - 5
ER -

Copy | Download

Author Information

Gikunju David Muriuki

Department of Mathematical Sciences, The Co-operative University of Kenya, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0001-0285-6692
Nyaga Lewis Namu

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

Contact Email

http://orcid.org/0009-0000-0004-9187

Download PDF

Abstract
Keywords
Document Sections
1. 1. Introduction
2. 2. Main Results
Show Full Outline
Abbreviations
Conflicts of Interest
References
Cite This Article
Author Information

Plain Text BibTeX RIS

APA Style

Muriuki, G. D., Namu, N. L. (2024). Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. American Journal of Applied Mathematics, 12(5), 175-182. https://doi.org/10.11648/j.ajam.20241205.17

Copy | Download

ACS Style

Muriuki, G. D.; Namu, N. L. Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Am. J. Appl. Math. 2024, 12(5), 175-182. doi: 10.11648/j.ajam.20241205.17

Copy | Download

AMA Style

Muriuki GD, Namu NL. Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets. Am J Appl Math. 2024;12(5):175-182. doi: 10.11648/j.ajam.20241205.17

Copy | Download

@article{10.11648/j.ajam.20241205.17,
  author = {Gikunju David Muriuki and Nyaga Lewis Namu},
  title = {Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets
},
  journal = {American Journal of Applied Mathematics},
  volume = {12},
  number = {5},
  pages = {175-182},
  doi = {10.11648/j.ajam.20241205.17},
  url = {https://doi.org/10.11648/j.ajam.20241205.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20241205.17},
  abstract = {The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all n ≥ 2 and the suborbits are length 1, (n-1), (n-1), (n-1), (n-1)2, (n-1)2, (n-1)2, (n-1)3. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all n > 2 and suborbital graphs of the group action are undirected. It is shown that graphs Γ2 and Γ3 are regular of degree n-1, graphs Γ4, Γ5 and Γ6 of degree (n-1)2 and graph Γ7 is regular of degree (n-1)3. The suborbital graphs Γi(i=1, 2,…, 6) are disconnected, with the number of connected components equal to n2 while suborbital graph Γ7 is connected for all n > 2.
},
 year = {2024}
}

Copy | Download

TY - JOUR
T1 - Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

VL - 12
IS - 5
ER -

Copy | Download