Ratio Mathematica Volume 40, 2021, pp. 247-257

Mixed and Non-mixed Normal Subgroups
of Dihedral Groups Using Conjugacy

classes

Vinod S∗

Biju G.S†

Abstract

In this paper, we characterize and compute the mixed and non-mixed
basis of Dihedral groups. Also, by computing the conjugacy classes,
we describe all the mixed and non-mixed normal subgroups of Dihe-
dral Groups.
Keywords: group; Dihedral group; mixed and non-mixed basis; nor-
mal subgroups; conjugacy classes;
2010 AMS subject classifications: 08A05. 1

∗Department of Mathematics, Government College for Women, Thiruvananthapuram, Kerala,
India; wenod76@gmail.com.
†Department of Mathematics, College of Engineering, Thiruvananthapuram-695016, Kerala,

India; gsbiju@cet.ac.in.
1Received on February 09th, 2021. Accepted on April 28th, 2021. Published on June 30th,

2021. doi: 10.23755/rm.v40i1.604. ISSN: 1592-7415. eISSN: 2282-8214. c©Vinod et al. This
paper is published under the CC-BY licence agreement.

247



Vinod S, Biju G.S

1 Introduction
There are many interesting functions from the family of Dihedral groups to set

of natural numbers. For the Dihedral group Dn of order 2n, Cavior [1975] proved
that the number of subgroups is d(n) + σ(n) where σ(n) is the sum of positive
divisors of n and d(n) denote number of positive divisors of n. For elemen-
tary facts about dihedral groups see Conrad [Retrieveda]. Conrad [Retrievedb]
describes the subgroups of Dn, including the normal subgroups. using characteri-
zation of dihedral groups in terms of generators and relations. Calugareanu [2004]
presents a formula for the total number of subgroups of a finite abelian group. In
Tărnăuceanu [2010] an arithmetic method is developed to count the number of
some types of subgroups of finite abelian groups.

Subgroups of groups of smaller sizes are widely studied because their group
properties can be easily verified and larger groups are usually studied in terms of
their subgroups (see Miller [1940]). In this paper we characterize and compute
the different basis of Dihedral groups. Also we describe all mixed and non-mixed
normal subgroups of Dihedral groups via conjugacy classes.

2 Notations and Basic Results
Most of the notations, definitions and results we mentioned here are standard

and can be found in Gallian [1994] and Dummit and Foote [2003]. For any given
natural number n denote:

d(n) = the number of positive divisors of n.
σ(n) = the sum of positive divisors of n.
ϕ(n) = the number of non- negative integers less than n and relatively

prime to n.

Also, the greatest common divisor of m and n is denoted by (m,n). Let G be a
group and a1,a2, . . . ,ap ∈ G. Then the subgroup generated by a1,a2, . . . ,ap is
denoted by < a1,a2, . . . ,ap >.

Definition 2.1. A group generated by two elements r and s with orders n and 2
such that srs−1 = r−1 is said to be the nth dihedral group and is denoted by Dn.

Theorem 2.1. For each divisor d of n, the group Zn has a unique subgroup of
order d, namely

〈n
d

〉
.

Theorem 2.2. For each divisor d of n, the group Zn has exactly ϕ(d) elements of
order d, namely {k

n

d
: 0 ≤ k ≤ d−1, (k,d) = 1}.

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Mixed and Non-mixed Normal Subgroups of Dihedral Groups Using Conjugacy
classes

Theorem 2.3. The number of subgroups of Zn is d(n), namely
〈n
d

〉
where d is a

divisor of n.

Theorem 2.4. Let G be a group generated by a and b such that an = e, b2 = e
and bab−1 = a−1. If the size of G is 2n then G is isomorphic to Dn.

By theorem 2.4, we make an abstract definition for dihedral groups.

Definition 2.2. For n ≥ 3, let Rn = {r0,r1, . . . ,rn−1} and Sn = {s0,s1, . . . ,sn−1}.
Define a binary operation on Gn = Rn ∪Sn by the following relations:

ri · rj = ri+j mod(n) ri ·sj = si+j mod(n)
si ·sj = ri−j mod(n) si · rj = si−j mod(n) for all 0 ≤ i,j ≤ n−1.

Then (Gn, ·) is a group of order 2n.

Note that in the group (Gn, ·), the identity element is r0, ri = rj if and only if
i = j mod(n), si = sj if and only if i = j mod(n), the inverse of ri is rn−i and
the inverse of si is si for all 0 ≤ i,j ≤ n − 1. It is also clear that ri1 = ri and
rj · s0 = sj for all 0 ≤ i,j ≤ n− 1. Since Gn is a group of order 2n and can be
generated by r1 and s0 such that:

rn1 = rn = r0, s
2
0 = r0 and s0r1s

−1
0 = s0r1s0 = s−1s0 = r−1 = rn−1 = r

−1
1 .

Therefore the group Gn is isomorphic to Dn =< r1,s0 >. The elements of Rn
are called rotations and that of Sn are called reflections. A subgroup of Dn which
contain both rotations and reflections is called a mixed subgroup and subgroups
contain rotations only is called non-mixed subgroup. From the group Dn, we have
the following.

Theorem 2.5. Rn is a subgroup of Dn and is isomorphic to Zn.

Theorem 2.6. If n is even, the number of elements of order 2 in Dn is n + 1,
namely {rn/2,sj : 0 ≤ j ≤ n−1}.

Theorem 2.7. If n is odd, the number of elements of order 2 in Dn is n, namely
{sj : 0 ≤ j ≤ n−1}.

Theorem 2.8. If d divide n and d 6= 2, the number of elements of order d in Dn is
ϕ(d) namely {rkn/d : 0 ≤ k ≤ d−1, (k,d) = 1}.

Theorem 2.9. If a and b are two elements in Dn, then < a,b >= {akbm : 0 ≤
k,m ≤ n−1}

Definition 2.3. Let G be a finite group. An element y ∈ G is said to be a conjugate
of x ∈ G iff y = gxg−1, for some g in G.

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Vinod S, Biju G.S

This relation conjugacy in a group G is an equivalence relation on G. The
equivalence class determined by the element x is denoted by cl(x). Thus cl(x) =
{gxg−1 : g ∈ G}. The summation,

∑
x∈G

|cl(x)|, where summation runs over one

element from each conjugacy class of x is called the class equation of G.

Definition 2.4. A subgroup H of the group G is said to be a normal subgroup if
ghg−1 ∈ H for all g ∈ G and h ∈ H.

A normal subgroup which contain rotations alone is called a non- mixed nor-
mal subgroup and normal subgroups which contains both reflections and rotations
is called mixed normal subgroup.

Theorem 2.10. Every normal subgroup is a union of conjugacy classes.

Theorem 2.11. Every subgroup of a cyclic normal subgroup of the group G is
also normal in G.

3 Subgroups of Dn
Theorem 3.1. The number of non-mixed subgroups of Dn is d(n), namely
{< rn/d >: d is a divisor of n}.

Proof. The non-mixed subgroups of Dn are subgroups of Rn. Since Rn is
isomorphic to Zn, for each divisor d of n, the group Rn has a unique subgroup of
order d, namely < rn/d >. Hence the number of non-mixed subgroups of Dn is
d(n), namely {< rn/d >: d is a divisor of n}. 2

Theorem 3.2. Every mixed subgroup of Dn has even order of which half of them
are rotation and half of them are reflection.

Proof. Let H be a mixed subgroup of Dn containing a reflection s. Let
A denote the set of rotations of H and B denote the set of all reflections of H.
Define a map ψ : A → B by ψ(r) = r · s for all r ∈ A. If sj is an element in B
then sj · s is an element of A and ψ(sj · s) = sjss = sj. Hence ψ is onto. Also
ψ(r) = ψ(r′) =⇒ rs = r′s =⇒ r = r′ and hence ψ is one-one. 2

Theorem 3.3. Every mixed subgroup of Dn is Dihedral.

Proof. Let H be a mixed subgroup of Dn. By theorem 3.2 , |H| = 2d for
some d and H ∩ Rn =< rn/d >. Since order of H is 2d and < rn/d > is its
subgroup of order d, we have H =< rn/d > ∪ < rn/d > s =< rn/d,s >, for
some s in H. Since (rn/d)d = ro,s2 = r0 and srn/ds−1 = (rn/d)−1, we have
H ≡ Dd and hence the proof. 2

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Mixed and Non-mixed Normal Subgroups of Dihedral Groups Using Conjugacy
classes

Corolary 3.1. If H is a mixed subgroup of Dn then,

1. |H| = 2d, for some d which divides n.

2. H ≡ Dn =< rn/d,s > for some s ∈ H.

Here we have a usual question: If d divides n, does there exist a subgroup of
order 2d? If it exists, how many?

Theorem 3.4. If d divides n, the number of mixed subgroups of order 2d is
n

d
.

Proof. By the corollary 3.1, it is clear that the mixed subgroups Dn of order
2d are {< rn/d,sj >: 0 ≤ j ≤ n− 1}, all of them need not be distinct. Suppose
< rn/d,si >=< rn/d,sj > for some 0 ≤ i,j ≤ n−1.

< rn/d,si > =< rn/d,sj >

⇐⇒ < rn/d > ∪ < rn/d > si =< rn/d > ∪ < rn/d > sj
⇐⇒ < rn/d > si =< rn/d > sj

⇐⇒ sis−1j ∈< rn/d >
⇐⇒ sis−1j = rkn/d for some 0 ≤ k ≤ d−1
⇐⇒ sisj = rkn/d
⇐⇒ ri−j = rkn/d

⇐⇒ i− j ≡
kn

d
mod(n) for some 0 ≤ k ≤ d−1

⇐⇒ d(i− j) ≡ 0mod(n)

⇐⇒ i− j ≡ 0mod
(n
d

)
⇐⇒ i ≡ j mod

(n
d

)
Hence the number of distinct mixed subgroups of order 2d in Dn is

n

d
, namely

{< rn/d,si >: 0 ≤ i <
n

d
}. 2

Theorem 3.5. The number of mixed subgroups of Dn is σ(n).

Proof. By theorem 3.4, the mixed subgroups of Dn is
∑
d/n

n

d
=
∑
d/n

d = σ(n).

They are ∪d/n{< rn/d,si >: 0 ≤ i ≤
n

d
−1}. 2

From theorem 3.1 and theorem3.5 we have,

Theorem 3.6. The number of subgroups of Dn is σ(n) + d(n).

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Vinod S, Biju G.S

Theorem 3.7. The number of abelian subgroups of Dn is d(n) +n if n is odd and
d(n) + n +

n

2
if n is even.

Proof. All non-mixed subgroups of Dn are cyclic and hence abelian. So by
theorem 3.1, there are d(n) non- mixed abelian subgroups for Dn. If n is odd, by
theorem 3.3 and corollary 3.1, the mixed abelian subgroups of Dn are of order 2
and hence there are n such subgroups. Thus if n is odd, the number of abelian
subgroups of Dn is d(n) + n. If n is even, by theorem 3.3 and corollary 3.1, the
mixed abelian subgroups of Dn are of order 2 and 4, and hence there are n +

n

2
such subgroups. Thus if n is even, the number of abelian subgroups of Dn is
d(n) + n +

n

2
. 2

Theorem 3.8. The number of cyclic subgroups of Dn is d(n) + n.

Proof. By theorem 3.1, the number of non-mixed cyclic subgroups of Dn is
d(n). Also by theorem 3.3 and corollary 3.1,the mixed cyclic subgroups of Dn is
n. Hence the number of cyclic subgroups of Dn is d(n) + n. 2

4 Basis of Dn
A basis of Dn which contain both rotation and reflection is called a mixed

basis and other basis is called non-mixed basis. By the definition 2.2, it is obvious
that two rotations cannot generate Dn. Hence non-mixed basis of Dn are basis
consisting of two reflections.

Theorem 4.1. For n ≥ 3, the number of mixed basis of Dn is nϕ(n).

Proof. Let sj(0 ≤ j ≤ n − 1) be a reflection in Dn. Then for any 0 ≤ i ≤
n−1,

< ri,sj > = {rmi s
t
j : 0 ≤ m,t ≤ n−1} ; by theorem 2.9

= {rmi sj, r
m
i r0 : 0 ≤ m ≤ n−1} ; since s

t
j = sj or r0

= {rmi sj, r
m
i : 0 ≤ m ≤ n−1}

= {rmi sj : 0 ≤ m ≤ n−1}∪{r
m
i : 0 ≤ m ≤ n−1}

=< ri > sj∪ < ri >= Dn if and only if (i,n) = 1

Hence corresponding to each reflection sj(0 ≤ j ≤ n− 1) there are ϕ(n) mixed
bases, namely {{sj,ri} : 0 ≤ i ≤ n−1 and (i,n) = 1}. So the number of mixed
basis for Dn(n ≥ 3) is nϕ(n). 2

Theorem 4.2. For n ≥ 3, the number of non-mixed basis of Dn is
nϕ(n)

2
.

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Mixed and Non-mixed Normal Subgroups of Dihedral Groups Using Conjugacy
classes

Proof. Since the dimension of Dn is 2, any basis of Dn contain exactly
two elements. The subgroup generated by two rotations always lies in Rn and
hence cannot form a basis. Therefore any non- mixed basis of Dn contain exactly
two reflections. : Let sj(0 ≤ j ≤ n − 1) be a reflection in Dn. Then for any
0 ≤ i ≤ n−1,

< si,sj > =< ri−jsj,sj >=< ri−j,sj >
∼= Dn if and only if i− j ≡ k mod(n) and (k,n) = 1

Hence corresponding to each reflection sj(0 ≤ j ≤ n − 1) there are ϕ(n) non-
mixed basis for Dn namely {{si+j,sj} : 0 ≤ i ≤ n − 1 and (i,n) = 1}. If
{si,sj} is a mixed basis corresponding to the reflection si, then it is also a ba-
sis corresponding to the reflection sj. Hence the number of non-mixed basis for

Dn(n ≥ 3) is
nϕ(n)

2
. 2

Theorem 4.3. For n ≥ 3, the number of different basis for Dn is
3n

2
ϕ(n).

Proof. The collection of all different bases of Dn(n ≥ 3) is the union
of all mixed and non-mixed bases. Hence the different bases of Dn(n ≥ 3) is
nϕ(n)

2
+ nϕ(n) =

3n

2
ϕ(n). 2

5 Congugacy classes of Dn
In this section we will compute all conjugacy classes and class equation of

Dihedral groups.

Theorem 5.1. If n is odd, the number of conjugacy classes in Dn is
n + 3

2
.

Proof. Let ri(0 ≤ i ≤ n−1) be a rotation in Dn. Then

cl(ri) = {rjrir−1j ,sjris
−1
j : 0 ≤ j ≤ n−1}

= {rjrir−j,sjrisj : 0 ≤ j ≤ n−1}
= {ri,sjrisj : 0 ≤ j ≤ n−1}
= {ri,sj−isj : 0 ≤ j ≤ n−1}
= {ri,r−i}

Since n is odd, ri = r−i if and only if i = 0. Therefore

cl(r0) = {r0} and cl(ri) = {ri,r−i}, a two element set, for all 1 ≤ i ≤ n−1.

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Vinod S, Biju G.S

Also,

cl(s0) = {rjs0r−1j ,sjs0s
−1
j : 0 ≤ j ≤ n−1}

= {rjs0r−1j ,sjs0sj : 0 ≤ j ≤ n−1}
= {rjs0r−j,sjs0sj : 0 ≤ j ≤ n−1}
= {s2j : 0 ≤ j ≤ n−1}
= {sj : 0 ≤ j ≤ n−1}, since n odd.

Hence, if n is odd, {{sj : 0 ≤ j ≤ n−1},{r0},{ri,r−i } : 1 ≤ i ≤ (n−1)/2}
are the conjugacy classes of Dn. Thus if n is odd, the number of conjugacy class

of Dn is
(n−1)

2
+ 2 =

(n + 3)

2
. 2

Corolary 5.1. The class equation of Dn(n odd ) is 1 + 2 + 2 + . . . + 2 + n = 2n,
the summation runs over (n−1)/2 times.

Theorem 5.2. If n is even, the number of conjugacy classes in Dn is
n + 6

2
.

Proof. Let ri(0 ≤ i ≤ n−1) be a rotation in Dn. Then

cl(ri) = {rjrir−1j ,sjris
−1
j : 0 ≤ j ≤ n−1} = {rjrir−j,sjrisj : 0 ≤ j ≤ n−1}

= {ri,sjrisj : 0 ≤ j ≤ n−1}
= {ri,sj−isj : 0 ≤ j ≤ n−1}
= {ri,r−i}

Since n is even ri = r−i if and only if i = 0 or
n

2
. Therefore

cl(r0) = {r0},cl(rn/2) = {rn/2} and cl(ri) = {ri,r−i}, a two element set, for all

1 ≤ i ≤ n−1 and i 6=
n

2
.

Also,

cl(s0) = {rjs0r−1j ,sjs0s
−1
j : 0 ≤ j ≤ n−1}

= {rjs0r−1j ,sjs0sj : 0 ≤ j ≤ n−1}
= {rjs0r−j,sjs0sj : 0 ≤ j ≤ n−1}
= {s2j : 0 ≤ j ≤ n/2−1}

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Mixed and Non-mixed Normal Subgroups of Dihedral Groups Using Conjugacy
classes

Again,

cl(s1) = {rjs1r−1j ,sjs1s
−1
j : 0 ≤ j ≤ n−1}

= {rjs1r−1j ,sjs1sj : 0 ≤ j ≤ n−1}
= {rjs1r−j,sjs1sj : 0 ≤ j ≤ n−1}
= {s2j+1 : 0 ≤ j ≤ n−1}
= {s2j+1 : 0 ≤ j ≤ n/2−1}

Hence, if n is even,{
{s2j : 0 ≤ j < n/2},{s2j+1 : 0 ≤ j < n/2},{r0},{rn/2},

{ri,r−i } : 1 ≤ i ≤ (n−2)/2
}

are the conjugacy classes of Dn. Thus if n is even, the number of conjugacy class

of Dn is
(n−2)

2
+ 4 =

(n + 6)

2
. 2

Corolary 5.2. The class equation of Dn(n even ) is 1 + 1 + 2 + 2 + . . . + 2 +
n/2 + n/2 = 2n, the summation runs over (n−2)/2 times.

Corolary 5.3. Each conjugacy class of Dn contains either rotations alone or re-
flections alone.

Corolary 5.4. The number of conjugacy classes of Dn which contain rotations

alone is
(n + 1)

2
if n is odd and

(n + 2)

2
if n is even.

Corolary 5.5. The number of conjugacy classes of Dn which contain reflec-

tions alone is 1, namely Dn, if n is odd and is 2, namely
{
{s2j : 0 ≤ j <

n/2},{s2j+1 : 0 ≤ j < n/2}
}

, if n is even.

6 Normal subgroups of Dn
In this section we will describe all mixed and non-mixed normal subgroups of

Dn.

Theorem 6.1. The number of non-mixed normal subgroups of Dn is d(n).

Proof. Since Rn is a cyclic normal subgroup of Dn, by theorem 2.11, the
non-mixed subgroups and non-mixed normal subgroup of Dn are same. Hence
the number of non-mixed normal subgroups of Dn is d(n). 2

255



Vinod S, Biju G.S

Theorem 6.2. The number of mixed normal subgroups of Dn is 1 if n odd and 3
if n even.

Proof. Since normal subgroups are union of conjugacy classes, a mixed
normal subgroup contain at least one conjugacy class having reflection. If n is odd,
there is only one conjugacy class having reflection, namely {sj : 0 ≤ j ≤ n−1}.
Therefore Dn is the only mixed normal subgroup of Dn if n is odd. If n even,
{s2j : 0 ≤ j < n/2} and {s2j+1 : 0 ≤ j < n/2} are the only conjugacy
classes having reflection. Therefore {s2j,r2j : 0 ≤ j < n/2},{s2j+1,r2j :
0 ≤ j < n/2} and Dn are the only mixed normal subgroups of Dn if n is even.
Therefore the number of mixed normal subgroups of Dn is 3 if n is even.

2

Corolary 6.1. The number of normal subgroups of Dn is d(n) + 1 if n odd and
d(n) + 3 if n even.

7 Conclusion
In this paper, it is proved that the number of mixed basis and non-mixed basis

for Dn(n ≥ 3) are nϕ(n) and
nϕ(n)

2
respectively, where ϕ(n)is the number of

non- negative integers less than n and relatively prime to n. Also it is shown that

the number of different bases for Dn(n ≥ 3) is
3n

2
ϕ(n). If n is odd, the number

of conjugacy classes in Dn is
n + 3

2
and if n is even, the number of conjugacy

classes in Dn is
n + 6

2
. Finally we have shown that the number of non-mixed

normal subgroups of Dn is d(n) and the number of mixed normal subgroups of
Dn is 1 if n odd and 3 if n even.

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Mixed and Non-mixed Normal Subgroups of Dihedral Groups Using Conjugacy
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