International Journal of Applied Sciences and Smart Technologies

Volume 3, Issue 1, pages 101–110

p-ISSN 2655-8564, e-ISSN 2685-9432

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Identity Graph of Finite Cyclic Groups

Maria Vianney Any Herawati1,*, Priscila Septinina Henryanti1,

Ricky Aditya1

1Department of Mathematics, Faculty of Science and Technology,

Sanata Dharma University, Yogyakarta, Indonesia
*Corresponding Author: any@usd.ac.id

(Received 26-03-2021; Revised 24-04-2021; Accepted 05-05-2021)

Abstract

This paper discusses how to express a finite group as a graph, specifically

about the identity graph of a cyclic group. The term chosen for the graph is

an identity graph, because it is the identity element of the group that holds

the key in forming the identity graph. Through the identity graph, it can be

seen which elements are inverse of themselves and other properties of the

group. We will look for the characteristics of identity graph of the finite

cyclic group, for both cases of odd and even order.

Keywords: Graph, identity graph, group, identity element.

1 Introduction

Mathematics as a science has several branches including abstract algebra and graph

theory [1-3]. The phrase of abstract algebra has been used since the early 20th century

to distinguish them from what is now more commonly referred to as elementary algebra,

which is the study of the rules of manipulation of algebraic formulas and expressions

involving real or complex variables and numbers [4-6]. Abstract algebra is a field of

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mathematics that studies algebraic structures, such as monoids, groups, rings, fields,

modules, etc. [3, 4].

Students often find it difficult to learn abstract structure such as group. Therefore,

some writers are looking for a way to represent a group with a diagram called a graph.

Graph theory is a branch of mathematics that has been studied and developed by

researchers. In its development, the application of graph theory is often found both in

mathematics itself and in other fields such as computer science, biology, chemistry and

in problems in human life such as transportation problems, installation of public

facilities, and traffic light management.

In this paper, graph theory will be applied in abstract algebra, especially to represent

groups in the form of a graph so that it can be visualized diagrammatically and studied

its properties through the graph of the group. The group discussed here is a finite group.

There are several previous articles that examine graphs formed from groups including

Cayley graphs, G-graphs, coprime graphs, and identity graphs of dihedral groups.

2 Methodology: Notations and Definitions

The method used is literature study with the initial step of forming an identity graph

of several cyclic groups then looking for general patterns of their properties, making

conjectures and proving them. Before going into those steps, in this section we will

discuss some basic concepts and definitions in group theory and graph theory.

Group Theory

These are some definitions in group theory [4] which will be used in the next section:

1. Group is a set with one binary operation on the set which fulfills associative

properties, has an identity element, and each member of the group has an inverse.

2. Order of a group is the number of its elements. A finite group is a group of finite

order. Let 𝑒 is the identity element of a finite group 𝐺. Order of an element a in 𝐺 is

defined as the smallest natural number 𝑛 such that 𝑎𝑛 = 𝑒.

3. Let 𝐺 be a group. A non-empty subset 𝐻 ⊆ 𝐺 is called a subgroup of 𝐺 if and only

if 𝐻 is also a group with the same operation defined in 𝐺 [4].

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4. If G is a group and 𝑎 ∈ 𝐺, then the set 〈𝑎〉 = {𝑎𝑛 ∶ 𝑛 ∈ 𝑍} is a subgroup of 𝐺, and

〈𝑎〉 is called the cyclic subgroup generated by 𝑎. Group 𝐺 is called 𝑎 cyclic group if

and only if 𝑎 ∈ 𝐺 exists, such that 𝐺 = 〈𝑎〉 [4].

Related to order of groups and order of elements, we have these two important theorems

in group theory [4]:

1. (Cauchy’s Theorem) Let 𝐺 be a finite group and p be a prime number. If 𝑝 divides

the order of 𝐺, then 𝐺 has an element of order 𝑝.

2. (Lagrange’s Theorem) If H is a subgroup of a finite group 𝐺, then order of 𝐻

divides order of 𝐺.

Graph Theory

Graph 𝐺 is a pair of finite sets (𝑉, 𝐸), written with the notation 𝐺 (𝑉, 𝐸), in which

case 𝑉 is a non-empty set of vertices and 𝐸 is a non-empty set of edges connecting a

pair of vertices or connect a vertex with the vertex itself. 𝐴 graph 𝐺 can be represented

by a diagram, each vertex of 𝐺 is represented by a dot or small circle while an edge

connecting two vertices is represented by a curve connecting the corresponding vertices

in the diagram.

3 Results and Discussion

In this section, we write our research results in terms of theorems and their proofs.

Some illustrations of graphs are also presented. First, we need to understand the concept

of identity graph of a group.

Definition [2]

Let G be a group. The identity graph of group 𝐺 is a graph with the elements of group 𝐺

as its vertices which satisfies these properties:

a) Two elements 𝑥, 𝑦 in group 𝐺 are connected by an edge if 𝑥𝑦 = 𝑒, with 𝑒 is the

identity element for group 𝐺.

b) Each element of 𝐺 is connected by an edge with the identity element 𝑒.

To develop the previous research, we shall examine the identity graph of finite cyclic

groups. There are two possibilities for the order of a finite cyclic group: it is an odd

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natural number, or it is an even natural number. The order may also be a prime number.

We will examine the case of odd prime order first.

Theorem 1 [2]

If 𝐺 = 〈𝑔 |𝑔𝑝 = 1, 𝑝 ≠ 2〉 is a cyclic group of the 𝑝th order, where p is prime, then the

identity graph formed by 𝐺 consists of (𝑝 − 1)/2 triangles.

Proof:

Let 𝐺 = 〈𝑔 | 𝑔^𝑝 = 1, 𝑝 ≠ 2〉 be a cyclic group of the 𝑝th order, where p is

prime. Then 𝐺 does not have a proper subgroup, according to Lagrange theorem.

Therefore, there is no element in 𝐺 having inverse which is itself; in other words, there

is no 𝑔𝑖 ∈ 𝐺 such that (𝑔𝑖 )
2

= 1. Suppose there is 𝑔𝑖 in 𝐺 such that (𝑔𝑖 )^2 = 1. Then,

𝐺 has a subgroup 𝐻 = {1, 𝑔𝑖 }. This contradicts with the fact that 𝐺 does not have a

proper subgroup. As a result, 𝐺 does not have element of the 2nd order. Using Cauchy

theorem with 𝑝 ≠ 2 then 𝐺 does not have element of the 2nd order.

For every(𝑔𝑖) in 𝐺 there is exactly one inverse of 𝑔𝑖 that is 𝑔𝑗 such that 𝑔𝑖

𝑔𝑗 = 1 with 𝑖 and 𝑗 are positive integers and 𝑖 ≠ 𝑗. Because 𝑔𝑖 𝑔𝑗 = 1 then 𝑔𝑖 𝑔𝑗 =

𝑔𝑖+𝑗 = 1 = 𝑔𝑝 such that 𝑝 = 𝑖 + 𝑗 which is equivalent to 𝑗 = 𝑝 − 𝑖. Consequently, we

can form the identity graph of 𝐺 as illustrated by Figure 1.

Figure 1. Identity graph of 𝐺 = 〈𝑔 | 𝑔𝑝 = 1, 𝑝 ≠ 2〉

𝑔
𝑝−1

2
+2

= 𝑔
𝑝+3

𝑔
𝑝−3

2 = 𝑔
𝑝+1

2
−2

𝑔
𝑝+1

𝑔
𝑝−1

𝑔𝑝−1

𝑔2

𝑔𝑝−2

𝑔

⋯

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We have seen a unique pattern of the identity graph of a finite cyclic group with odd

prime order. We skip the case of even prime order since the only even prime number is

2, and the identity graph of group of order 2 looks too trivial and not so interesting. Now

we look at more general case of finite cyclic groups of odd order.

Theorem 2 [2]

If 𝐺 is a cyclic group with an odd order, then 𝐺 has the identity graph 𝐺𝑖 which can be

formed by triangles without a unique edge.

Proof:

Let 𝐺 = 〈𝑔 |𝑔𝑛 = 1〉 with n is an odd integer is a cyclic group with multiplication

operation and is of the nth order. The elements of 𝐺 are {1, 𝑔, 𝑔2, 𝑔3, 𝑔4, ⋯ , 𝑔𝑛−1 }. We

shall use the Cayley table given by Table 1 to show results of the operation for each

element of 𝐺.

Table 1. Cayley table for Theorem 2.

1 𝑔 𝑔2 𝑔3 ⋯ 𝑔𝑛−3 𝑔𝑛−2 𝑔𝑛−1

1 1 𝑔 𝑔2 𝑔3 ⋯ 𝑔𝑛−3 𝑔𝑛−2 𝑔𝑛−1

𝑔 𝑔 𝑔2 𝑔3 𝑔4 ⋯ 𝑔𝑛−2 𝑔𝑛−1 1

𝑔2 𝑔2 𝑔3 𝑔4 𝑔5 ⋯ 𝑔𝑛−1 1 𝑔

𝑔3 𝑔3 𝑔4 𝑔5 𝑔6 ⋯ 1 𝑔 𝑔2

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮

𝑔𝑛−3 𝑔𝑛−3 𝑔𝑛−2 𝑔𝑛−1 1 ⋯ 𝑔𝑛−6 𝑔𝑛−5 𝑔𝑛−4

𝑔𝑛−2 𝑔𝑛−2 𝑔𝑛−1 1 𝑔 ⋯ 𝑔𝑛−5 𝑔𝑛−4 𝑔𝑛−3

𝑔𝑛−1 𝑔𝑛−1 1 𝑔 𝑔2 ⋯ 𝑔𝑛−4 𝑔𝑛−3 𝑔𝑛−2

From the Cayley table above (Table 1) we can see that 𝑔𝑔𝑛−1 = 1, 𝑔2 𝑔𝑛−2 =

1, … , 𝑔𝑛−1𝑔 = 1. Thus, for any non-identity element g, we have 𝑔𝑘 𝑔𝑛−𝑘 = 1, 𝑓𝑜𝑟 𝑘 ∈

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𝑍+ and 𝑘 < 𝑛. From the definition of identity graph part (𝑖𝑖𝑖), for any elements 𝑎, 𝑏 ∈

𝐺, 𝑎 ≠ 𝑏, 𝑎 ≠ 𝑒, 𝑏 ≠ 𝑒, there exists an edge which connects 𝑎 to 𝑏 if and only if 𝑎𝑏 =

𝑏𝑎 = 𝑒. Therefore, there exists an edge which connects 𝑔𝑘𝑡𝑜 𝑔𝑛−𝑘, and since for every

𝑎 ∈ 𝐺, 𝑎 ≠ 𝑒 there exists an edge connecting 𝑎 to 𝑒, then from the definition of identity

graph part (𝑖𝑖), there exists an edge from 𝑔𝑘 to 1. These will form a triangle connecting

1, 𝑔𝑘 and 𝑔𝑛−𝑘. Moreover, since n is an odd number, then 𝑛 = 2𝑥 + 1 for any 𝑥 ∈

𝑍+, and the number of non-identity elements in 𝐺 is even. So, those non-identity

elements can be partitioned into two sets with same cardinality, where the inverse of an

element in one set is in the other set and vice versa. Therefore, there is no single edge in

identity graph of 𝐺.

Then the identity graph which corresponds with 𝐺 is shown in Figure 2.

Figure 2. Identity graph of group 𝐺 = {𝑔 | 𝑔𝑛 = 1}, 𝑛 is an odd number.

Based on previous theorems, we can conclude a characterization of the identity graph of

cyclic group of odd order in the following Theorem 3.

Theorem 3 [2]

If 𝐺 = 〈𝑔|𝑔𝑛 = 1〉 is a cyclic group of order n, where n is an odd number, then the

identity graph 𝐺𝑖 of 𝐺 is formed by (𝑛 − 1)/2 triangles.

Proof:

𝑔

𝑔2

𝑔3

𝑔𝑛−1

𝑔𝑛−2

𝑔𝑛−3

⋯

𝑔𝑘

𝑔𝑛−𝑘

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This can be proved using Theorem 1 and Figure 1. Since the number of non-identity

elements is even and those elements forms (n-1)/2 couples, where each couple is inverse

to each other, then there will be (n-1)/2 triangles.

Similar with the result in the odd order case, we can obtain a characterization of the

identity graph of cyclic group of even order case in the following Theorem 4. The proof

is also using similar principle with the odd order case.

Theorem 4 [2]

If 𝐺 = 〈𝑔|𝑔𝑚 = 1〉 is a cyclic group of order m where m is an even number, then its

identity graph 𝐺𝑖 has (𝑚 − 2)/2 triangle and a single edge.

Proof:

Table 2. Cayley table for Theorem 4.

1 𝑔 𝑔2 𝑔3 ⋯ 𝑔
𝑚

2 ⋯ 𝑔
𝑚−3 𝑔𝑚−2 𝑔𝑚−1

1 1 𝑔 𝑔2 𝑔3 ⋯ 𝑔
𝑚

2 ⋯ 𝑔
𝑚−3 𝑔𝑚−2 𝑔𝑚−1

𝑔 𝑔 𝑔2 𝑔3 𝑔4 ⋯ 𝑔
𝑚

2
+1

⋯ 𝑔
𝑚−2 𝑔𝑚−1 1

𝑔2 𝑔2 𝑔3 𝑔4 𝑔5 ⋯ 𝑔
𝑚

2
+2

⋯ 𝑔
𝑚−1 1 𝑔

𝑔3 𝑔3 𝑔4 𝑔5 𝑔6 ⋯ 𝑔
𝑚

2
+3

⋯ 1 𝑔 𝑔
2

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮

𝑔
𝑚

2 𝑔
𝑚

2
+1

𝑔
𝑚

2
+2

𝑔
𝑚

2
+3

⋯ 1 ⋯ 𝑔
3𝑚

2
−3

𝑔
3𝑚

2
−2

𝑔
3𝑚

2
−1

⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮

𝑔𝑚−3 𝑔𝑚−3 𝑔𝑚−2 𝑔𝑚−1 1 ⋯ 𝑔
3𝑚

2
−3

⋯ 𝑔
𝑚−6 𝑔𝑚−5 𝑔𝑚−4

𝑔𝑚−2 𝑔𝑚−2 𝑔𝑚−1 1 𝑔 ⋯ 𝑔
3𝑚

2
−2

⋯ 𝑔
𝑚−5 𝑔𝑚−4 𝑔𝑚−3

𝑔𝑚−1 𝑔𝑚−1 1 𝑔 𝑔2 ⋯ 𝑔
3𝑚

2
−1

⋯ 𝑔
𝑚−4 𝑔𝑚−3 𝑔𝑚−2

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Let 𝐺 = 〈𝑔 |𝑔𝑚 = 1〉, m is an even number, be a cyclic multiplicative group and its

order is m. Elements of 𝐺 are {1, 𝑔, 𝑔2, 𝑔3, 𝑔4, ⋯ , 𝑔𝑚−1 }. We will use the Cayley table

given by Table 2 to show the operations between elements in 𝐺.

From the Cayley table above (Table 2) we can see that 𝑔𝑔𝑚−1 = 1, 𝑔2 𝑔𝑚−2 =

1, … , 𝑔𝑚−1 𝑔 = 1. However, for 𝑔
𝑚

2 we have something different, that is 𝑔
𝑚

2 𝑔
𝑚

2 = 1.

Thus, for any non-identity elements other than 𝑔
𝑚

2 we have 𝑔𝑘 𝑔𝑚−𝑘 = 1, for 𝑘 ∈

𝑍+ and 𝑘 < 𝑚. From the definition of identity graph part (iii), for any elements 𝑎, 𝑏 ∈

𝐺, 𝑎 ≠ 𝑏, 𝑎 ≠ 𝑒, 𝑏 ≠ 𝑒, vertices 𝑎 and 𝑏 are adjacent if and only if 𝑎𝑏 = 𝑏𝑎 = 𝑒. This

means the vertex 𝑔^𝑘 is adjacent with vertex 𝑔𝑚−𝑘 and since for each 𝑎 ∈ 𝐺, 𝑎 ≠ 𝑒

vertex 𝑎 is adjacent with 𝑒 based on definition of identity graph part (ii), then there exists

an edge from 𝑔𝑘 to 1. These form a triangle connecting 1, 𝑔𝑘 dan 𝑔^(𝑚 − 𝑘). In other

side, for 𝑔
𝑚

2 there will be only one edge connecting it, that is the edge which connects

𝑔
𝑚

2 with 1. Moreover, since m is even, then 𝑚 = 2𝑥 for 𝑥 ∈ 𝑍+ and the number of

elements which is not the identity and not 𝑔
𝑚

2 in 𝐺 is even. Therefore, there will be

(𝑚 − 2)/2 triangles and a single edge. The identity graph 𝐺𝑖 which corresponds with 𝐺

is given in Figure 3.

Figure 3. Identity graph of group 𝐺 = 〈𝑔| 𝑔𝑚 = 1〉, 𝑛 is an even number.

𝑔

𝑔2

𝑔3

𝑔𝑚−1

𝑔𝑚−2

𝑔𝑚−3

𝑔𝑘

𝑔𝑚−𝑘

𝑔
𝑚

⋯

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We see that in the even order case, there exists exactly one single edge that does not form

a triangle. This happens because in a finite cyclic group of even order, there exists

exactly one element of order 2.

4 Conclusion

From what we have discussed, there are some conclusions about identity graph of

finite cyclic groups as following:

a. Identity graph of a group is a way to represent the relations between elements of a

group in a graph. In identity graph of a group, the identity element of a group is

connected with any other elements and each non-identity element is connected with

its inverse.

b. For cyclic group of odd order 𝑛, its identity graph consists of (𝑛 − 1)/2 triangles.

There is no single edge in this case since in such group there is no element of

order 2.

c. For cyclic group of even order 𝑚, its identity graph consists of (𝑚 − 2)/2 triangles

and a single edge. The single edge connects the identity element to the only element

of such group that has order 2.

References

[1] A. Bretto, A. Faisant, L. Gilibert, G-graphs: A new representation of group, Journal

of Symbolic Computation, 42, 549-560, 2007.

[2] W. B. V. Kandasamy, F. Smarandache, Groups as Graphs. Slantina: Editura CuArt.

2009.

[3] S. Lovett, Abstract Algebra. Boca Raton: CRC Press. 2016.

[4] C. C. Miller, Essentials of Modern Algebra. Dulles, VA: Mercury Learning and

Information, 2013.

[5] R. Rajkumar, P. Devi, Coprime Graph of Subgroups of a Group,

https://www.semanticscholar.org

[6] M. U. Sherman-Bennett, On Groups and Their Graphs, MA: Bard College, 2016.

https://www.semanticscholar.org/

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