A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

Alolaiyan, Hanan; Razaq, Abdul; Yousaf, Awais; Zahra, Rida

doi:https://doi.org/10.1155/2021/3905425

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Research Article | Open Access

Volume 2021 | Article ID 3905425 | https://doi.org/10.1155/2021/3905425

A Comprehensive Overview on the Formation of Homomorphic Copies in Coset Graphs for the Modular Group

Hanan Alolaiyan,¹Abdul Razaq,²Awais Yousaf,³and Rida Zahra⁴

Academic Editor: Kenan Yildirim

Received09 Aug 2021

Accepted03 Sept 2021

Published21 Sept 2021

Abstract

This work deals with the well-known group-theoretic graphs called coset graphs for the modular group G and its applications. The group action of G on real quadratic fields forms infinite coset graphs. These graphs are made up of closed paths. When M acts on the finite field Z_p, the coset graph appears through the contraction of the vertices of these infinite graphs. Thus, finite coset graphs are composed of homomorphic copies of closed paths in infinite coset graphs. In this work, we have presented a comprehensive overview of the formation of homomorphic copies.

1. Introduction

The study of groups via their actions has been a central theme in group theory, since the completion of the classification of simple groups in the 1980s. This most commonly takes the form of actions on vector spaces and similar commutative objects or on more elementary combinatorial objects. It is not an exaggeration to say that the modular group G (see [1–4]) is the single most important infinite discrete group, through its myriad connections with number theory, geometry, and topology. There is a long and venerable history of studying its actions, particularly on finite sets, which goes back to before the turn of the twentieth century. The modular group G has two generators f and , where f: x ⟶ −1/x and g: x ⟶ x − 1/x are linear fractional transformations. The finite presentation of G is <f, : f² = = 1>. It means that it is a free product of C₂ and C₃. The linear fractional transformation h: x ⟶ 1/x extends G to G′ because it inverts f and ; that is, h² = (fh)² = (gh)² = 1. Thus, extended modular group G′ has three generators f, , and h and its finite presentation is <f, , h: f² = = h² = (fh)² = (gh)² = 1>.

Graph theory has applications in various branches of mathematics [5, 6]. Several topological and algebraic structures can be studied in a more effective way by using graphs. Graphical techniques are specifically utilized to investigate the finitely generated groups. The graphs prove to be an effective and simple method to solve many mathematical problems [7–9].

The use of graphs to represent group actions has a venerable history. Cayley [8] published the first work on this topic. Mathematicians like Coxeter [9], Burnside [10], Stothers [11], Everitt [12], Conder [13], Whitehead [14], and others provided pioneering works on graphical representations of groups. The action of a modular group on certain objects can be represented by a certain type of graphs, called coset graphs. These were introduced by Higman in 1978. Later, in 1983, Mushtaq [15] laid their foundation. These graphs consist of triangles connected to each other. The edges of triangles are permuted anticlockwise to represent by . Each vertex of a triangle is connected by f to another vertex of the triangle (which may be the same triangle).

Moreover, the vertices of the coset graph that are fixed by f and are represented by heavy dots. Since (gh)² = 1 implies hgh⁻¹ = ⁻¹, h turns around the direction of the triangles like reflection. Thus, we do not introduce h-edges in coset graphs, so that they remain simple.

The action of G′ on finite field Z_p is not possible because f maps 0 to ∞. Thus, we add ∞ to Z_p in order to make the action possible.

Example 1. Let us consider the action of modular group on . The permutation representations f, , and h areThe corresponding coset graph is shown in Figure 1.

Definition 1. Let and be two coset graphs; then is a homomorphic copy of if(i)order of is less than the order of , that is, ,(ii)u is a vertex in such that (u) x = u for some , and then there exists some vertex such that .Let be a real quadratic irrational number; then , where m is a square-free natural number and . In [16], Mushtaq studied the group action of G on real quadratic fields and showed that the corresponding coset graphs are infinite. Figure 2 shows a small patch of these graphs.
Due to the emergence of infinite graphs, the action of on through coset graphs is not easy to study. Therefore, the action of G on becomes important. The coset graphs for are homomorphic copies of the infinite graphs for , where , for any natural number n. For example, the coset graph shown in Figure 1 is the homomorphic copy of the coset graph for because .
For further details about coset graphs, we refer the readers to [17–22].
The main contributions of this paper are as follows:(1)A thorough study on the formation of homomorphic copies of coset graphs is presented(2)We have developed a formula to compute all homomorphic copies of the closed path of rank 4

2. Closed Paths in Coset Graph

Definition 2. A closed path in a coset graph containing a vertex fixed by , where is called a closed path of rank k. It is denoted by . In [23], it has been proved that the rank of closed paths is always even.

Remark 1. Let and . Then vertex is fixed by
Suppose that and are any two vertices in a closed path C, such that and . Let ; then also maps to . Clearly, and are the only possible paths to travel from to . By contraction of vertices and , we mean and merge to form a node such that is fixed by both and . This can be done by making a closed path containing u such that and then by applying on such that ends at . Consequently, a graph is evolved, which is a homomorphic copy of . Note that, in addition to and , there are some other pairs of vertices in , which also compose by contraction. In fact, during the formation of by contracting and , some more pairs also get contracted. How many are they? The following theorems help to calculate this number.

Theorem 1 (see [24]). Let a homomorphic copy of be formed by contracting its vertices and . Then is obtainable also by contracting the pair for some .

Theorem 2 (see [24]). The number of pairs to obtain is equal to the number of elements , such that and lie in .

Example 2. Consider a closed path (see Figure 3) containing a vertex which is a fixed point of . Thus, it is a rank two closed path, denoted by .
Figure 4 represents the homomorphic copy of generated through contraction of vertices and .

3. Formation of Homomorphic Copies through Contraction of Vertices

The coset graphs are made up of closed paths. The vertices of infinite graphs are contracted in a specific manner to evolve finite coset graphs. Therefore, a question arises: how many distinct homomorphic copies can be created by contracting all pairs in a closed path? In this work, we have developed a technique to find all homomorphic copies of the closed paths , where and , in coset graphs. Diagrammatically , where and , is shown in Figure 5.

In the remaining part of the paper, we denote the closed path , where and , by . Throughout this paper, the mirror image of any homomorphic copy is denoted by . If , where or , then let . If fixes any vertex , then the vertex fixed by is .

Remark 2. Since (gh)² = 1 implies hgh⁻¹ = , h turns around the direction of the triangles like reflection. If is obtained by contracting vertices and of any closed path C, then the mirror image of can be created by contracting and . It should be noted that and do not need to lie in the same closed path C. From Figure 5, we havefor , , , and . Thus, for each vertex u in , there exists a vertex in .

Remark 3. Some homomorphic copies have symmetry about the vertical axis; that is, they have the same orientations as those of their mirror images. In other words, they are mirror images of themselves. The homomorphic copy of any circuit C having a vertex fixed by has a symmetry about vertical axis if and only if contains a vertex fixed by ,.

3.1. Proposed Scheme

Since has number of vertices, the total number of pairs in is . We contract a pair of vertices of such that a homomorphic copy is obtained. By using Theorem 2, we find all pairs of vertices in , which form ; let those be n in numbers. Now, we have two possibilities:(i)If by contracting and vertices and are not contracted, then does not possess a vertical symmetry. Therefore, consumes n more pairs of vertices of .(ii)If and , are contracted all together, then has a symmetry about vertical axis. Therefore, does not consume any pair. Thus, has n pairs of vertices.

Next, we contract at one of the remaining pairs and the process continues until all pairs are exhausted.

Let . First, we contract vertex with vertices and the following result is obtained.

Theorem 3. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Proof. Let be the set of homomorphic copies of obtained by contracting with . In Figure 5, it can be seen that and are the possible paths between and . Therefore, vertex in is fixed by and . It is also clear from Figure 5 that is the set of elements of the modular group such that, for all , both and lie in . Since , by Theorem 2, there are pairs in to form .
Next, we show that all homomorphic copies of in are different and no copy of these is a mirror image of another.
Let ; then is evolved by contracting and , whereas is obtained by contracting and . Now if and only if there exists an element in such that and . One can see that only maps to itself, but .
Now suppose that ; then there must exist some in which sends to and to . But does not contain such element. This means that all diagrams in are distinct. Thus, and there are pairs of vertices to create .
Now we check how many diagrams in have a symmetry about vertical axis. For this, let ; then contains an element such thatThis is possible only if ; in this case, we have such that and . So, we conclude that only has a symmetry about vertical axis; that is, and its mirror image have the same orientations, and all other homomorphic copies in do not possess a vertical symmetry. Hence there are pairs to form .
We obtain all the results by using the same technique, so, from now onwards, instead of providing proofs of the theorems, we will present tables, which give the necessary information of the family of homomorphic copies evolved.
Let . We contract the pair and and obtain the following result. By using Theorem 1, one can see that these vertices are not contracted in Theorem 3.

Theorem 4. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Table 1 shows the complete information of the family of homomorphic copies . The information provided in Table 1 can be verified by the same technique used in the proof of Theorem 3.

Let us contract with , where . By using Theorem 1, it can be easily verified that these pairs of vertices are not utilized in the previous theorems.

Theorem 5. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All details of the family of homomorphic copies have been provided in Table 2.

Next, we contract with and formulate the following Theorem.

Theorem 6. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

For complete information of , see Table 3.

Now we contract vertex with vertices , where , and acquire the following result.

Theorem 7. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Table 4 provides all information regarding homomorphic copies , evolved in Theorem 7.

The following theorem is evolved by contracting with , where .

Theorem 8. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

The complete information of the family of homomorphic copies obtained in Theorem 8 is given in Table 5.

The next theorem is obtained by contracting with , where .

Theorem 9. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All details of the generated homomorphic copies are given in Table 6.

Now we contract vertex with vertices for all .

Theorem 10. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

The complete information of the family of homomorphic copies is provided in Table 7.

In the following theorem, is contracted with .

Theorem 11. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All details of homomorphic copies can be found in Table 8.

Let us now contract vertex with vertices , where .

Theorem 12. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All information related to the homomorphic copies created in Theorem 12 is given in Table 9.

Let us contract vertex with vertices , where . Consequently, we have the following theorem.

Theorem 13. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Table 10 completely describes the family of homomorphic copies evolved in Theorem 11.

Suppose that and . Let us contract with to obtain Theorem 14.

Theorem 14. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All information of the homomorphic copies evolved in Theorem 14 is provided in Table 11.

The following theorem emerges as a result of contracting with , where .

Theorem 15. If vertex is contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Table 12 shows the complete information of all homomorphic copies created in Theorem 15.

Suppose that and . Let us contract with and obtain the following results.

Theorem 16. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All necessary details of the family of homomorphic copies evolved in this process are provided in Table 13.

Let , where . The following results have been constructed by contracting with .

Theorem 17. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

See Table 14 for complete information of the homomorphic copies.

Recall that and . Let , and we obtain the following theorem by contracting with .

Theorem 18. If vertices are contracted with vertex , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All information related to the homomorphic copies created in Theorem 18 is provided in Table 15.

Now we contract with , where and , so that Theorem 19 is evolved.

Theorem 19. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

For complete details about the homomorphic copies evolved in Theorem 19, see Table 16.

Let . The following theorem is obtained by contracting with .

Theorem 20. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All important information regarding is provided in Table 17.

Recall that , and let and . In Theorem 21, we contract vertices with vertices to create homomorphic copies .

Theorem 21. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All information about the family of homomorphic copies evolved in Theorem 21 is given in Table 18.

Now, let us contract vertices with , where and .

Theorem 22. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

Table 19 provides all information regarding .

Recall that and let and . The next theorem is obtained by contracting with

Theorem 23. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

All details of the generated homomorphic copies are given in Table 20.

Now suppose that . The following results have been constructed by contracting with .

Theorem 24. If vertices are contracted with vertices , then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

See Table 21 for complete details of the evolved homomorphic copies.

Now, we contract with , where

Theorem 25. If vertices are contracted with vertices then distinct homomorphic copies of are obtained. Furthermore, there are pairs of vertices for these homomorphic copies.

The complete information of the family of homomorphic copies obtained in Theorem 25 is given in Table 22.

Suppose that . Let us contract and and obtain the following results.

Theorem 26. If vertices are contracted with vertices , then distinct homomorphic copies of are evolved. Furthermore, there are pairs of vertices for these homomorphic copies.

All information of the family of homomorphic copies evolved in Theorem 26 is provided in Table 23.

Lastly, we contract 4 pairs of vertices to evolve 4 homomorphic copies.

Theorem 27. (i)Let be the homomorphic copy of obtained by contracting and . Then there are pairs of vertices for .(ii)Let be the homomorphic copy obtained by contracting and . Then there are pairs of vertices for .(iii))Let be the homomorphic copy obtained by contracting and . Then there are pairs of vertices to generate (iv)Let be the homomorphic copy obtained by contracting and Then there are pairs of vertices for .Let . We are now in a position to prove the main theorem of this paper.

Theorem 28. There are numbers of distinct homomorphic copies of by contracting all the pairs in .

Proof. Let us contract the following pairs of vertices:(i) and (ii) and (iii) and (iv) and (v) and (vi) and (vii) and (viii) and (ix) and (x) and (xi) and (xii) and (xiii) and (xiv) and (xv) and (xvi) and (xvii) and (xviii) and (xix) and (xx) and (xxi) and (xxii) and (xxiii) and (xxiv) and (xxv) and (xxvi) and (xxvii) and (xxviii) and Then, by Theorems 3 to 27, we obtain the setof homomorphic copies of and there arepairs to form . is the total number of pairs in andThus, there arenumbers of distinct homomorphic copies obtained by contracting all pairs in .

4. Conclusion

In this paper, we have highlighted the significance of homomorphic copies of coset graphs. We show how the bigger coset graphs get contracted to transform into their homomorphic copies. As a sample, we consider the closed path of rank 4; that is, , where and , in coset graphs. We prove that the total numbers of homomorphic copies of are

It is hoped that the ideas presented in this study will also help to generate the homomorphic copies of other types of graphs for various purposes.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this article.

Acknowledgments

This paper was supported by Researchers Supporting Project (no. RSP-2021/317), King Saud University, Riyadh, Saudi Arabia.

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Copyright © 2021 Hanan Alolaiyan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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