401 

© 2024 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

PISSN: 1609-4042, EISSN: 2521-3407 

 IHJPAS. 2024, 37(4) 

 
 

Analysing the Result Involution Graph of the Group J3 

 

Ahmed Arkan Meteab1 and  Ali Abd Aubad2,*   
1,2Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq. 

*Corresponding Author.  

 

Received: 12 April 2023                        Accepted: 18 May 2023                 Published: 20 October 2024 

doi.org/10.30526/37.4.3405 

 
 Abstract  

        Assume that G be a finite group and let I(G) be the set of the involution elements in G. The 

result involution graph denoted by , Γ𝐺
𝑅𝐼, is an undirected simple graph having the elements 

of G as a vertex set. Moreover, two vertices in Γ𝐺
𝑅𝐼are connected by an edge if they are 

distinct and their product belong to I(G). The objective of this work is to investigate the result 

involution graph for the Janko group J3. In this paper we compute different result involution 

graph features, such as the radius, the diameter, the clique number, and the girth. Furthermore, 

the connectedness of the result involution graph is determined. All of the steps needed for 

analyzing the result involution graph were carried out using the computational technique along 

with theoretical support.  

Keywords: Janko Groups, Result Involution Graph, Connectedness, Girth.  

 

1. Introduction 

One of the most important techniques for examining group structure is possibly the action 

of  a group on a graph. The numerous and diverse graphs that have been analyzed for various 

groups are highlighted by the substantial citations in [1-14]. If an element of a group has order 2, 

it is said to be involution. Involutions of groups significantly affect the way it is structured. For 

instance, the categorization of finite simple groups heavily depends on the study of involutions. 

An S3-involution graph for a group G was first introduced by Devillers and Giudici in [15] as a 

graph with a G-classes of involution as a vertex set, where two vertices are adjacent if they generate 

an S3-subgroup in a certain G-class. They looked at the structure  of S3-involution graph and 

general characteristics for the group PSL (2,q) when q is greater than 3. Define I(G) as a set of the 

involution elements in a finite group G. The result involution graph is known as the simple 

undirected graph with vertex set being the elements of G with two vertices x,y G are adjacent if 

x≠y and xy I(G), and is denoted by Γ𝐺
𝑅𝐼. Jund and Salih reported the result involution graph and 

its attributes for the first time in [16]. In their analysis, they demonstrated that the graphs  Γ𝑆𝑛
𝑅𝐼 and 

 Γ𝐴𝑛
𝑅𝐼 are connected graphs for n>4 and have a maximum diameter, radius, and girth of 3 and 3. 

Also, they provide some really helpful qualities for the result involution graphs of the quaternion 

group and the dihedral group. Nevertheless, Aubad and Salih looked at the structure of the 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0001-4526-3790
mailto:ahmed.arkan1203a@sc.uobaghdad.edu.iq
https://orcid.org/0000-0002-4764-2946
mailto:ali.abd@sc.uobaghdad.edu.iq


IHJPAS. 2024, 37(4 ) 

402 
 

  

 

involution graphs that were produced for the whole list of the Mathieu sporadic simple groups  as 

mentioned in [17]. Furthermore, the result involution graphs for the Conway group Co3 are 

analyzed with full information in [18]. 

 Examining the result involution graph for the Janko group J3, is the aim of this article. Together 

with the connectedness of the result involution graph, we show many graph properties including 

the radius, diameter, clique number, and girth. The paper is set up like follows: We offer some 

observations and graph notations for the result involution graph in section 2. A number of results 

about  Γ𝐽3
𝑅𝐼 are provided in section 3. We draw our concluding conclusions and make suggestions 

for more study in section 4. 

 

2.  Preliminary 

We begin the section by defining a few concepts and outlining several facts that will be 

important to the thesis later on. First let G be a finite group moving forward. Let sIG=|I(G)|  be 

the size of the set I(G), where I(G) be the set of the involution elements in G as mentioned in the 

introduction. We begin with the following formula to determine how many edges there are 

in the result involution graph: 

 

Proposition 2.1 [17]: Let G be a finite group, and let F represent the number of elements 

of order 4 in G. Then the number of edges in the result involution graph is calculated using 

the following formula ½ (𝒔𝐈𝑮 |𝐺| − 𝐹). 

 

We should notice that if G is a finite group with large number of elements, then the  Γ𝐺
𝑅𝐼 

has a huge vertex set. The result  involution graph of G is then exceedingly difficult to handle. 

To resolve this issue, we use the resize graph idea, which define as follows: 

Definition 2.2 [17]: Assume that G be a finite group, the  resize graph of G, Γ𝐺
𝑅𝑆, has a vertex 

set the whole G-conjugacy classes with two vertices ( G-conjugacy classes) X,Y  G,  are 

adjacent  if X≠Y and  there are v1 X and v2 Y such that v1, v2 are adjacent  vertices in  Γ𝐺
𝑅𝐼.   

The following results displays a relationship between a result involution graph and a resize 

graph: 

Proposition 2.3 [17]: Let G be a finite group. Then the connectivity of the  graphs  Γ𝐺
𝑅𝐼 and 

Γ𝐺
𝑅𝑆  are isomorphic graphs. 

These findings will be illustrated by the example that follows. We will mostly employ GAP 

[19] and YAGS [20] as a computational approach in the following example. 

Example 2.4: Let G be an alternating group of degree 4. Then the result involution graph Γ𝐴4
𝑅𝐼 

contains the following vertices:{e,(2,3,4), (2,4,3), (1,2)(3,4), (1,2,3), (1,2,4), (1,3,2), (1,3,4), 

(1,3)(2,4), (1,4,2), (1,4,3), (1,4)(2,3)}.Then we can conclude the figure below that the graph Γ𝐴4
𝑅𝐼  

is disconnected with two components. One of these components has four vertices labels by the 

numbers {1,9,4,12} and the subgraph induced with this vertex is the compete graph K4. On the 

other hand, the subgraph induced by the vertices labels by {2,3,5,6,7,8,10,11}. This graph is 3-

regular graph with dimeter and  radius of size 3. Moreover, clique number of size 2 and girth 4. 

This information can be seen the following Figure: 



IHJPAS. 2024, 37(4 ) 

403 
 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.The Structure of the Result Involution Graph Γ𝐺
𝑅𝐼 , GA4. 

 

On the other hand, the resize graph  of G, Γ𝐴4
𝑅𝑆 has vertex set present as {1A,2A,3A,3B}. In the 

next figure we can see that resize graph of A4 is (1,1)-biregular disconnected graph has vertex 

set can be split into two disjoint vertices can be labeled by X={1,2} and Y={3,4}:  

 

 

 

 

 

 

 

 

 

Figure 2. The Structure of the Resize Graph Γ𝐺
𝑅𝑆, GA4. 

3.  Results  

In this section we will look at the structure of the result involution graph for the Janko 

group J3. To achieve the goal of the study, we will use GAP and the Online Atlas [21]. We 

should note that the group J3 has only one class of involution namely 2A with size equal to 

26163. There are several involutions to take into account as a consequence. Hence, in order 

to arrive at our observations, we will investigate the resizing graph Γ𝐽3
𝑅𝑆. 

3.1 The Structures of 𝚪𝑱𝟑
𝑹𝑺 

The Online Atlas  reveals that J3 has different  21 conjugacy classes. Therefore , the resizing 

graph Γ𝐽3
𝑅𝑆 has order 42. Furthermore, the  vertex set of Γ𝐽3

𝑅𝑆 could now be stated in the following 

manner:{1A ,  2A ,  3A ,  3B ,  4A ,  5A ,  5B ,  6A ,  8A ,  9A ,  9B ,  9C ,  10A ,  10B ,  12A ,  

15A ,  15B ,  17A ,  17B ,19A , 19B}. 

In addition, the vertex degree sequence is shown below: 

{1, 18, 14, 17, 18, 19, 19, 18, 19, 18, 18, 18, 19, 19, 19, 19,19, 19, 19, 18, 18} 

 



IHJPAS. 2024, 37(4 ) 

404 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3. The Structure of the Resize Graph Γ𝐺
𝑅𝑆, GJ3. 

According to the data in Figure 3, Γ𝐽3
𝑅𝑆 is connected and has the following characteristics: 

 

Table 1. Properties of the Resize Graph Γ𝐽3
𝑅𝑆 . 

Γ𝐺
𝑅𝑆  E(Γ𝐺

𝑅𝑆) Girth Clique Number Radius Diameter 

GJ3 183 3 17 2 3 

3.2 The Edges Set of Result Involution Graphs 

In this section, we provide comprehensive information about the edges set of the result 

involution graph Γ𝐽3
𝑅𝐼 for the Janko group J3. The data concern the number of edges connecting any 

two G-Conjugacy elements. The findings will be crucial in assessing the construct of the  result 

involution graph of J3. We will use GAP and the Online Atlas to get the intended result. We should 

be informed of that, in the elements of the class 5A there are an edges with the remaining J3-

conjugacy classes except the identity element. This can be seen in the following set: 

{2A(33488640),3A(25116480),3B(167443200),4A(418608000),5A(598609440),5B(157396608

0),6A(1783270080),8A(5425159680),9A(1557221760),9B(1557221760),9C(1557221760),10A(

4651572096),10B(4470733440),12A(3717239040),15A(3265142400),15B(2662346880),17A(2

561880960),17B(2561880960),19A(2310716160),19B(2310716160)}. 

In addition, we can find 26163 edges between the elements of the  class 2A. 

3.3 The Structures of 𝚪𝑱𝟑
𝑹𝑰 

For the Janko group J3, the structures of the result involution graph are fully described in 

the following theorem: 

Theorem 3.1: The result involution graph for the Janko group J3 is connected with diameter 3, 

radius 2,  and girth 3. 



IHJPAS. 2024, 37(4 ) 

405 
 

Proof: The resizing graph of J3 is connected using Table 1. As a result, by utilizing  Proposition 

2.3, we are able to determine how Γ𝐽3
𝑅𝐼 is connected. In addition, between the elements of the  

class 2A there are 26163 edges and these elements adjacent with the identity element of J3. 

Hence, we have a cycle with a length of 3 in the graph Γ𝐽3
𝑅𝐼. Therefore, the result involution 

graph has girth 3 in this case. Furthermore, The class 5A vertices and the identity element are not 

adjacent. But, during the classes 2A, these vertices were connected to the identity. As a result, in 

this scenario, the radius is 2. Moreover, the diameter is 3 due to the other vertices maximum 

distance being 3. 

We provide the following significant corollaries in light of the preceding finding: 

Corollary 3.2: Suppose that G be the Janko groups J3. Then for any distinct elements x,y  

G, one of the following are holds: 

i- Their product is an involution element. 

ii-  There exist an certain z  G, satisfy its product with x and y  produce an  involution 

elements. 

iii-  There are an elements z,w  G, satisfy  (xz), (zw)  and (wy) are involution elements. 

iv- J3 has three distinct components, any one of them  has an involution product with the 

others. 

Proof: By applying  the outcomes of Theorem 3.1 we have the diameter 2 and the  radius 2 for 

the  result involution graph of G . Therefore, when  x and y  have distance  one  then their product 

is an involution element so (i) is hold.  If  the distance between  x and y equal 2, then one can find 

z in G satisfying (ii). Finally, to prove (iii) the distance between x and y equal to 3, implies  there 

are such  z,w  G satisfy the conditions of (iii). Finally, the result involution graph of G with 

girth 3. As a result, the characterization of the result involution graph normally leads to the proof 

of iv.  

 

4. Conclusion 

This study analyses the result involution graph for the Janko group J3. the computational 

method used to determine certain graph properties. Detailed information about the resize graph, 

for example, as well as the radius, diameter and circumference. The results of this study can be 

applied to the study of more complicated simple groups, including monster groups, pariah groups 

and Lie-type exceptional groups. 

 

Acknowledgment 

        The authors would like to thank the University of Baghdad (https://uobaghdad.edu.iq/), 

Department of Mathematics in the College of Sciences for their motivation and support. 

 

Conflict of Interest  

The authors declare that they have no conflicts of interest. 

 

Funding 

No funding. 

 

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