SOME RESULTS OF NON-COPRIME GRAPH OF THE

DIHEDRAL GROUP 𝑫𝟐𝒏 FOR 𝒏 A PRIME POWER

Wahyu Ulyafandhie Misuki1,a), I Gede Adhitya Wisnu Wardhana2, *)Ni Wayan

Switrayni3,b) and Irwansyah4,c)

1,2,3,4
Algebra Research Group, Universitas Mataram, Jl Majapahit No.62, Mataram, 83125, Indonesia
e-mail: a)wahyu.misuki@unram.ac.id; b)niwayan.switrayni@unram.ac.id; c)irw@unram.ac.id
*)
Corresponding author: adhitya.wardhana@unram.ac.id

Abstract.A graph of a finite group 𝐺 whose vertices are all elements of 𝐺 except the identity element, and edges
defined as (𝑢, 𝑣) ∈ 𝐸(𝐺) if and only if (|𝑢|, |𝑣|) ≠ 1 is called a non-coprime graph of 𝐺 and denoted by ̅̅̅ Γ𝐺 . In this
paper we give some properties of non-coprime graphs of a dihedral group 𝐷2𝑛 , when 𝑛 is a prime power. One main
result of this paper shows that ̅̅̅
Γ𝐺 is either a complete graph or can be partitioned into two complete graphs.

INTRODUCTION

The non-coprime graph was first introduced by Mansori [3] who gave some of its characterizations. The
non-coprime graph is a dual represetation of the coprime graph that introduced by Ma [5]. Other authors also
studied graph representation of groups especially on coprime graph such as on dihedral group by Gazir [2] and on
group of integer modulo by Juliana [4].

In this paper we use the results of Gazir [2] on the coprime graph of dihedral group to find its dual
representation graph, the non-coprime graph of the dihedral group 𝐷2𝑛 , where 𝑛 is a prime power.

Definition 1. [2] A graph is an ordered set(𝑉, 𝐸) comprising:

i. The set 𝑉 is non-empty set of vertices
ii. The set 𝐸 is edge set of a pair vertices, 𝐸 ⊆ {(𝑣, 𝑤) ∈ 𝑉 2 }

Two vertices 𝑣𝑖, 𝑣𝑗 are said to be neighbors or adjacent if (𝑣𝑖, 𝑣𝑗 ) ∈ 𝐸. A graph is called an undirected
graph if (𝑥, 𝑦) = (𝑦, 𝑥) for every (𝑥, 𝑦) ∈ 𝐸, and called a simple graph if every edge (𝑥, 𝑦) ∈ 𝐸 is unique and
𝑥 ≠ 𝑦. In this paper we only consider an undirected simple graph.

Definition 2. [2] An undirected simple graph 𝐺 = (𝑉, 𝐸) is called a complete graph if for every 𝑥, 𝑦 ∈ 𝑉, then
(𝑥, 𝑦) ∈ 𝐸.

A mathematical system with a binary operation is called a group if it satisfies four conditions, namely
closure, associativity, identity and invertibility.

Definition 3. [1] A nonempty set 𝐺 is said to be a group if in 𝐺 there is defined a binary operation, called the
product and denoted by ∗ such that, for any 𝑎, 𝑏, 𝑐 ∈ 𝐺 then
i. 𝑎, 𝑏 ∈ 𝐺 implies that 𝑎 ∗ 𝑏 ∈ 𝐺 (closed).
ii. 𝑎 ∗ (𝑏 ∗ 𝑐) = (𝑎 ∗ 𝑏) ∗ 𝑐 (associative law).
iii. There exists an element 𝑒 ∈ 𝐺 such that 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎 for all 𝑎 ∈ 𝐺 (the existence of an identity
element).
iv. For every 𝑎 ∈ 𝐺 there exists an element 𝑎 −1 ∈ 𝐺 such that 𝑎 ∗ 𝑎−1 = 𝑎 −1 ∗ 𝑎 = 𝑒 (the existence of
inverse).

If (𝐺.∗) is a group that satisfies a commutative property, that is for any 𝑎, 𝑏 ∈ 𝐺, we have 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎,
then (𝐺.∗) is called a commutative group or an abelian group. A non-empty subset 𝐻 of a group 𝐺 is said to be a
subgroup of 𝐺 if 𝐻 is a group under the same operation on 𝐺. If |𝐻| is finite, it is easy to check that whether 𝐻 is
a subgroup or not. But if |𝐻| is infinite, then we can check whether 𝐻 is a subgroup or not using the following
theorem.

Theorem 1. [1] Let 𝐻 be a non-empty subset of a group 𝐺. Subset 𝐻 is a subgroup of 𝐺 if and only if 𝑎𝑏 −1 ∈ 𝐻,
for any 𝑎, 𝑏 ∈ 𝐻.

A way to represent a group into a graph is by describing it by the order of every element of the group.
The order of a group’s element is given by the following definition.

Definition 4. [1] Suppose (𝐺.∗) is any group. Let 𝑎 be any element of 𝐺. The smallest positive integer 𝑚 that
satisfies 𝑎𝑚 = 𝑒 is said as an order of 𝑎, and denoted as |𝑎| = 𝑚.

One of the most interesting groups is dihedral group, which is a group of symmetries of a regular polygon
consisting of rotations and reflections. Dihedral groups are playing an important role in group theory, geometry,
and chemistry.

Definition 5. [1] The dihedral group with order 2𝑛, denoted by 𝐷2𝑛 is the set:
𝐷2𝑛 = {𝑒, 𝑎, 𝑎2 , … , 𝑎𝑛−1 , 𝑏, 𝑎𝑏, 𝑎2 𝑏, … , 𝑎𝑛−1 𝑏|𝑎𝑛 = 𝑏 2 = 𝑒, 𝑎 −1 = 𝑏𝑎𝑏}for n ≥ 3.

By definition, we can find the order of every element of the dihedral group, depending on 𝑛. For any
natural number 𝑛, there always exists an element with order 2.

Theorem 2. [1] Let 𝐷2𝑛 = {𝑒, 𝑎, 𝑎2 , … , 𝑎𝑛−1 , 𝑏, 𝑎𝑏, 𝑎2 𝑏, … , 𝑎𝑛−1 𝑏|𝑎𝑛 = 𝑏 2 = 𝑒, 𝑎 −1 = 𝑏𝑎𝑏}, then|𝑏| = |𝑎𝑏| =
⋯ = |𝑎𝑛−1 𝑏| = 2

Gazir and Wardhana [1] found the characterizations of subgroup of a dihedral group in the following
Theorems:

Theorem 3. [1] Let 𝐷2𝑛 be the dihedral group with 𝑛 ≥ 3. If 𝑆 = {𝑒, 𝑎, 𝑎2 ,⋅⋅⋅, 𝑎𝑛−1 } ⊆ 𝐷2𝑛 then 𝑆 is a nontrivial
subgroup of 𝐷2𝑛 .

Theorem 4. [1] Let 𝐷2𝑛 be the dihedral group with 𝑛 ≥ 3. If 𝑆 = {𝑒, 𝑎𝑖 𝑏} ⊆ 𝐷2𝑛 where 𝑖 = 0,1,2,⋅⋅⋅, 𝑛 − 1 then
𝑆 is a non-trivial subgroup of 𝐷2𝑛 .

Theorem 5. [1] Let 𝐷2𝑛 be the dihedral group with 𝑛 ≥ 3. If n is composite where𝑛 = 𝑝1 𝑝2 ⋅⋅⋅ 𝑝𝑘 then 𝑆 =
{𝑒, 𝑎 𝑝𝑖 , 𝑎2𝑝𝑖 ,⋅⋅⋅ , 𝑎𝑛−𝑝𝑖 } ⊆ 𝐷2𝑛 is a non-trivial subgroup of 𝐷2𝑛 .

Mansori [3] gave the definition of non coprime graph of a group based on the order of every element of
the group. We denote that the order of element 𝑥 of a group as |𝑥| (look at Def 4).

Definition 6. [2] Let 𝐺be a finite group. Non-coprime graph of group 𝐺, denoted by ̅̅̅ Γ𝐺 is a graph with its vertices
consist of 𝐺 − {𝑒} and two different vertices 𝑢,𝑣are said to be adjacent if(|𝑢 |, |𝑣|) ≠ 1.

MAIN RESULTS

The non-coprime graph of 𝐷2𝑛 is a complete graph or can be partitioned into two complete graphs
whenever 𝑛 is a prime power.
 If 𝑛 is a power of even prime then the non-coprime graph of 𝐷2𝑛 is a complete graph, as shown in the
following theorem.

Theorem 6. Let 𝐷2𝑛 be the dihedral group. If 𝑛 = 2𝑚 for some 𝑚 ∈ ℕ, then ̅̅̅̅̅
Γ𝐷2𝑛 is a complete graph.

Proof. Since 𝑛 = 2𝑚 , then we have the order of 𝐷2𝑛 is 2𝑚+1 , hence the order of every non-identity element of
𝐷2𝑛 must be divided by 2.Then for any non-identity 𝑥, 𝑦 ∈ 𝐷2𝑛 we have (|𝑥|, |𝑦|) = 2𝑘 for some 𝑘 ∈ ℕ, hence
Γ𝐷2𝑛 is a complete graph. 
𝑥, 𝑦 are neighbors, then ̅̅̅̅̅

If 𝑛 is a power of an odd prime then the non-coprime graph of 𝐷2𝑛 can be partitioned into two complete
graphs.

Theorem 7. Let 𝐷2𝑛 be the dihedral group. If 𝑛 = 𝑝𝑚 for some 𝑚 ∈ ℕ and 𝑝 is an odd prime. Then ̅̅̅̅̅
Γ𝐷2𝑛 can be
partitioned into two disjoint complete graphs.
𝑚 𝑚
Proof. Let split 𝐷2𝑛 − {𝑒} into two disjoint sets 𝐺1 = {𝑎, 𝑎2 , … , 𝑎𝑝 −1 } and 𝐺2 = {𝑏, 𝑎𝑏, … , 𝑎𝑝 −1 𝑏}. We have
𝑝 divides the order of any 𝑥 ∈ 𝐺1 and 2 divides the order of any 𝑦 ∈ 𝐺2 since 𝒙 and 𝒚 are rotation and reflection
elements, respectively. So we have every two elements in 𝐺𝑖 are neigbors for 𝑖 = 1,2. Since 𝑝 is odd prime
then for any 𝑥 ∈ 𝐺1 and 𝑦 ∈ 𝐺2 , we have (|𝑥|, |𝑦|) = 1. Hence 𝑥 and 𝑦 cannot be neighbors, Therefore ̅̅̅̅̅Γ𝐷2𝑛 can
be partitioned into two disjoint complete graphs.

Subgroups of dihedral groups can be grouped into two types, that are trivial subgroups and non-trivial
subgroups. Obviously, the graph from a trivial subgroup of 𝐷2𝑛 satisfies the previous theorem.
These are all non-trivial subgroups of dihedral groups according to Gazir and Wardhana [1].
1. 𝑆 = {𝑒, 𝑎, 𝑎2 ,⋅⋅⋅, 𝑎 𝑛−1 }
2. 𝑆 = {𝑒, 𝑎𝑖 𝑏} where 𝑖 = 0,1,2,⋅⋅⋅, 𝑛 − 1
𝑘 𝑘 𝑘
3. 𝑆 = {𝑒, 𝑎𝑝𝑖 , 𝑎2𝑝𝑖 ,⋅⋅⋅, 𝑎𝑛−𝑝𝑖 } where 𝑛 = 𝑝1 1 𝑝2 2 ⋅⋅⋅ 𝑝𝑚𝑚
{𝑡} {𝑡} {𝑡} {𝑡}
{∏{𝑖=1} 𝑝𝑗𝑖 } {𝑛−∏{𝑖=1} 𝑝𝑗𝑖 } {𝑞+∏{𝑖=1} 𝑝𝑗𝑖 } {𝑞+𝑛−∏{𝑖=1} 𝑝𝑗𝑖 }
4. 𝑆 = {𝑒, 𝑎 ,⋅⋅⋅, 𝑎 , 𝑎𝑞 𝑏, 𝑎 𝑏,⋅⋅⋅, 𝑎 }, where 1 ≤ 𝑡 ≤ 𝑚 and 0 ≤ 𝑞 ≤
𝑛 − 1.

It is easy to check that when 𝑛 is a prime then all the non-trivial subgroups are only the first two
subgroups. In general, the non-coprime graph of any subgroup of 𝐷2𝑛 is either a trivial graph or a complete
graph.
Theorem 8. Let 𝑆 be a non trivial subgroup of dihedral group 𝐷2𝑛 . If 𝑛 = 𝑝𝑚 then the non-coprime graph of
𝑆 is a trivial graph or a complete graph or can be partitioned into two complete graphs.
Proof. Obviously for 𝑆 = {𝑒, 𝑎𝑖 𝑏}where 𝑖 = 0,1,2,⋅⋅⋅, 𝑛 − 1, Γ̅𝑆 is a trivial graph.
So we have three cases left.

Case 1 : 𝑆 = {𝑒, 𝑎, 𝑎2 ,⋅⋅⋅, 𝑎𝑛−1 }

The order of any non-identity element of 𝑆 must be divided by p, hence every non identity element of 𝑆 must be
neighbors. Then Γ̅𝑆 is a complete graph.
𝑖 𝑖 𝑖
Case 2 : 𝑆 = {𝑒, 𝑎𝑝 , 𝑎2𝑝 ,⋅⋅⋅, 𝑎𝑛−𝑝 } where 𝑖 = 1,2,⋅⋅⋅ , 𝑚.
Similar to case 1, we have the order of any non-identity element of 𝑆 must be divided by p, then we can conclude
that Γ̅𝑆 is a complete graph.
{𝑡} {𝑡} {𝑡} {𝑡}
{∏{𝑖=1} 𝑝𝑗 } {𝑛−∏{𝑖=1} 𝑝𝑗 } {𝑞+∏{𝑖=1} 𝑝𝑗 } {𝑞+𝑛−∏{𝑖=1} 𝑝𝑗 }
Case 3 : 𝑆 = {𝑒, 𝑎 𝑖 ,⋅⋅⋅, 𝑎 𝑖 , 𝑎𝑞 𝑏, 𝑎 𝑖 𝑏,⋅⋅⋅, 𝑎 𝑖 } , 1 ≤ 𝑡 ≤ 𝑚 and 0 ≤
𝑞 ≤ 𝑛 − 1.
If 𝑝 = 2, obviously the order of every non-identity element of 𝑆 must be divided by 2, hence we have Γ̅𝑆 is a
complete graph. If 𝑝 is an odd prime then the order of non-identity element of elements of 𝑆 is either 2 or divided
by 𝑝. Hence we can partition Γ̅𝑆 into two complete graphs.
 CONCLUSIONS

Given a dihedral group 𝐷2𝑛 with 𝑛 is a prime power, then the non-coprime graph of 𝐷2𝑛 is always a
complete graph or can be partitioned into two complete graphs. The same case happened to any subgroups of
𝐷2𝑛 .

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