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Research Article | Open Access

Volume 2021 |Article ID 6669459 | https://doi.org/10.1155/2021/6669459

Muhammad Aamir, Muhammad Awais Yousaf, Abdul Razaq, "Number of Distinct Homomorphic Images in Coset Diagrams", Journal of Mathematics, vol. 2021, Article ID 6669459, 39 pages, 2021. https://doi.org/10.1155/2021/6669459

# Number of Distinct Homomorphic Images in Coset Diagrams

Accepted28 Mar 2021
Published02 Aug 2021

#### Abstract

The representation of the action of on in a graphical format is labeled as coset diagram. These finite graphs are acquired by the contraction of the circuits in infinite coset diagrams. A circuit in a coset diagram is a closed path of edges and triangles. If one vertex of the circuit is fixed by, then this circuit is titled to be a length- circuit, denoted by. In this manuscript, we consider a circuit of length 6 as with vertical axis of symmetry, that is, . Let and be the homomorphic images of acquired by contracting the vertices and , respectively, then it is not necessary that and are different. In this study, we will find the total number of distinct homomorphic images of by contracting its all pairs of vertices with the condition The homomorphic images are obtained in this way having versatile applications in coding theory and cryptography. One can attain maximum nonlinearity factor using this in the encryption process.

#### 1. Introduction

It is prominent that the finite presentation is known as the modular group generated by the linear fractional transformations . In , Akbas discussed suborbitalgraphs for the modular group by showing that these graphs contains no circuit if and only if it contains no triangles. If we insert an extra generator with and, another group is emerged, denoted as , an extension of with the finite presentation as

In 1978, Professor Graham Higman propounds an unfamiliar type of a graph, titled as coset diagram, which presents the action of on , where is a finite field and shows a prime power. In 1983, this foundation is laid by Qaisar Mushtaq . Small triangles are proposed for the cycle , such that permutes the vertices of triangles in the opposite direction of rotation of clock and an edge is attached to any two vertices that are interchanged by . Heavy dots represent the fixed points of and . Note that equals , which means reverses the triangle orientation proposed for the cycle . For that reason, the diagram need not to be made more perplexing by interjecting edges.

A coset diagram (subdiagram) is said to be a homomorphic image of the coset diagram (subdiagram) if and only if with, where , there exist a vertex in such that .

Coset diagrams obtained from the action of over are infinite graphs , where . These diagrams are not easy to study because they are infinite. Thence, coset diagrams are considered as important for the action of on because this action presents finite graphs. The number is an expression of the number , where These finite coset diagrams are the homomorphic images of the coset diagrams for , where for some .

To explain more, coset diagram in Figure 1 illustrates the action on by with permutation representations , and by , , and, respectively, as

Thus, gives that the coset diagram is a homomorphic image of the coset diagram for .

Coset diagrams obtained from the action of on have some attractive narrative. In , the quadratic irrational numbers are classified by taking prime modulus that proved helpful in investigating the modular group action on the real quadratic field. The number is called the conjugate of , where and are integers and is a fixed number from , which is not a perfect square. is said to be an ambiguous number , if the sign of is different from the sign of . is said to be a totally negative (positive) if and both have the same signs. For a fixed , the number of ambiguous numbers of the form is finite and that segment of the coset diagram attained by the ambiguous numbers forms a closed path (circuit) and it is the only closed path in -orbit . With the help of coset diagram, Anna Torstensson not only described the applications to study the finitely presented group but also discussed the one-relator quotients of the modular group .

A closed path of triangles and edges in a coset diagram is called a circuit. In a coset diagram, a circuit is said to be a length- circuit, denoted by , if its one vertex is fixed by

Alternatively, it means that one vertex of the triangles lies outside of the circuit and one vertex of the triangles lies inside of the circuit and likewise. Since is a cycle, so it does not matter if one vertex of the triangles lies inside of the circuit and one vertex of the triangles lies outside of the circuit and likewise. Note that is always even.

The circuit of the type is termed as a periodic circuit with period of length .

Note 1. By we mean the collection of vertices lies on the circuit .

Let be any two vertices fixed by the words and , that is, and . Suppose is the word that maps to , then Note that and are the only two paths that assign to . Now, by contraction of the pair of vertices and , we mean that and melt together to become one node such that . As a result of this contraction, a closed path is created that contains the vertex fixed by and . This closed path is the homomorphic image of the circuit . It is important to note that and is not the only pair of contraction in that creates homomorphic image . There are also many pairs of contraction other than and that create the same homomorphic image . The following theorems proved in  will help us to find the total number of such contracted pairs that produce the same homomorphic image of .

Theorem 1. Let the vertices and in are contracted and a homomorphic image of is evolved, then is also obtainable if the vertices and in are contracted.

Theorem 2. If and are contracted to obtain , then during this process number of pairs are contracted all together, where is the collection of words such that and are contained by .

Example 1. Let us contract the vertices and from the circuit (Figure 2) and acquire a homomorphic (Figure 3) of the circuit . Thus, and are the two possible paths between and that are fixing the vertex in .
Let be the family of words such that for all implies and lie on the circuit , then . Then by Theorem 2, the cardinality of implies that there are 9 pairs of vertices contracted to generate the homomorphic image .
Note that the cardinality of does not give the total number of contracted pairs to generate the homomorphic image . In the following, we will discuss the process to find the total number of contracted pairs to generate .
Let denote itself as the mirror image of . Thus, the permutation ensures that the coset diagram is symmetric along the vertical axis. This implies will assuredly occur.
If is a word, then If the word fixes the vertex , then the vertex is fixed by .
A homomorphic image has a symmetry with respect to vertical axis if and only if by contracting and , the vertices and are also contracted.

Remark 1. In coset diagrams, reverses the orientation of the triangles representing the three cycles of (as reflection does). So corresponding to each vertex fixed by the pair , , there is a vertex in (mirror image of ) such that is a fixed point of , . In other words, it is created by contracting and . There are certain ’s which have a vertical symmetry and so have the same orientations as those of their mirror images. The homomorphic image of a circuit , which has a vertex fixed by the pair , , has the same orientation as that of its mirror image if and only if there is a vertex in such that .

##### 1.1. Counting the Number of Pairs of Contracting Vertices of a Homomorphic Image

Let be a homomorphic image of the circuit acquired by the contraction of pair of vertices and of . Then by Theorem 2, has number of pairs of vertices. To find the total number of pairs of vertices, one should follow the following steps.

To know how many total pairs of contracting vertices are there, special precaution must be taken.(1)If by contracting and to create , the pair of vertices and are not contracted, then has different orientation from its mirror image So, there are number of more pairs of vertices for the mirror image of .(2)If by contracting and to create , the pair of vertices and are also contracted, then has the same orientation as that of its mirror image So, in this case, has number of pairs of contracted vertices.

Consider a circuit of length 6 as (Figure 4) with vertical axis of symmetry, that is, . Suppose . The coset diagrams are composed of circuits. The vertices of the circuits in infinite diagrams are contracted in a certain way, and a finite coset diagram evolves. It is therefore necessary to ask how many distinct homomorphic images are obtained if we contract all the pairs of vertices of the circuit ? We not only give the answer to this question for a circuit but also mention those pairs of vertices which are “important”. There is no need to contract the pairs which are not mentioned as “important”. If we contract those, we obtain a homomorphic image, which we have already obtained by contracting “important” pairs.

Note 2. It is clear from Figure 4 that(1)The mirror image of the vertex is , that is, (2)The vertex is fixed by the word (3)The vertex is fixed by the word , where and

Lemma 1. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .

Proof. Let (Figure 5) be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in , where and are fixed by the words and . It is easy to verify that and are the possible paths between and (Figure 4). This implies that for each , the homomorphic image has a vertex fixed by and .
Thus,is the family of elements in such that contains the vertices and . This gives that the cardinality of , that is, is the number of contracted pairs of vertices to produce the homomorphic image (Theorem 2). For , let and be any two elements of , then the number of triangles in and are not equal (Figure 5). This implies that all the elements in are different and no one is the mirror image of the other. This further forms the result as . Thence, the number of contracted pairs of vertices of to create all the elements of is .
From Figure 5, it is also clear that no element of except has vertical axis of symmetry. So, is the only homomorphic image whose orientation is not different from its mirror image and all the remaining elements of have different orientations from their mirror images. Hence, there arepairs of contracted vertices to produce all the homomorphic images in .

Lemma 2. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .

Let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in . Suppose shows itself as the remainder of . Then, graphically we make four partitions of as follows:(i) (Figure 6(a))(ii) (Figure 6(b))(iii) and (Figure 6(c))(iv) and (Figure 6(d))

From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Let .

Lemma 3. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices that create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .

For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 79 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself.

This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 4. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .

Let be the collection of homomorphic images of acquired by the contraction of the vertices with the vertex in . Figure 10 presents graphically. From all the homomorphic images presented in Figure 10, it is not intricated to check that every one is the mirror image of itself.

This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 5. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the remainder of , then graphically, we make four partitions of as follows:(i) (Figure 11(a))(ii) (Figure 11(b))(iii) and (Figure 11(c))(iv) and (Figure 11(d))

From all the homomorphic images presented in these figures, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 6. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Graphically, we make two partitions of as follows:(i) (Figure 12(a))(ii) (Figure 12(b))From all the homomorphic images presented in Figures 12(a) and 12(b), it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.
Let.

Lemma 7. If we contract the vertices , with the vertex in the circuit , then for each there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
For a fix value of , let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figures 1315 present graphically. From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 8. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 16 presents graphically. From all the homomorphic images presented in Figure 16, it is not intricated to check that no one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 9. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the remainder of , then graphically we make four partitions of as follows:(i) (Figure 17(a))(ii) (Figure 17(b))(iii) and (Figure 17(c))(iv) and (Figure 17(d))From all the homomorphic images presented in these figures, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 10. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Graphically, we make two partitions of as follows:(i) (Figure 18(a))(ii) (Figure 18(b))From all the homomorphic images presented in these figures, it is not intricated to check that no one is the mirror image of itself except . This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 11. If we contract the vertices , with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Figure 19 presents graphically. From all the homomorphic images presented in Figure 19, it is not intricated to check that every one is the mirror image of itself. This lemma can be proved by using the same procedure as that for Lemma 1.

Lemma 12. If we contract the vertices, with the vertex in the circuit , then there arise distinct homomorphic images of and pairs of contracted vertices create each homomorphic image. Moreover, the number of total pairs of contracted vertices to generate all and their mirror homomorphic images of is .
Let be the collection of homomorphic images of acquired by the contraction of the vertices , with the vertex in . Suppose shows itself as the reminder of , then graphically we make four partitions of as follows:(i) (Figure 20(a))(ii)