SYMMETRIC GROUP

A PROJECT REPORT

Submitted by

BRISTI PATRA

UID: 22013121019

DEPARTMENT OF MATHEMATICS

BANKURA CHRISTIAN COLLEGE

for

B.Sc. SEMESTER - VI EXAMINATION, 2024 – 25

in
MATHEMATICS (HONOURS)
Of

BANKURA UNIVERSITY

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 TITLE OF THE PROJECT REPORT

Name of the Student: Bristi Patra

UID : 22013121019

Registration No. : 00387 of 2022 - 23

Semester : VI

Academic Session : 2024 – 25

Course Type : Honours [CBCS (NEW)]

Subject : MATHEMATICS

Course Title: Dissertation on any topic of Mathematics (Symmetric Group)

Course Code : SH/MTH/604/DSE – 4

Course ID : 62127

Name of the Examination: B.Sc. Semester - VI Examination, 2024 – 25

Name of the Supervisor: Dr. Arup Mukhopadhyay

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DEPARTMENT OF BANKURA CHRISTIAN COLLEGE
MATHEMATICS (Estd: 1903)

Re-accredited with Grade ‘A’ (4th Cycle) by NAAC in 2025

College with Potential for Excellence (2nd Phase)
PO+Dist: Bankura, West Bengal, INDIA, PIN: 722101
Phone No. 03242-250924
Website: www.bankurachristiancollege.in
email: principal@bankurachristiancollege.in

PROJECT COMPLETION CERTIFICATE

To Whom It May Concern

This is to certify that Bristi Patra (UID: 22013121019, Registration No.: 00387 of 2022-23) of

Department of Mathematics, Bankura Christian College, Bankura, has successfully carried out this

project work entitled “Symmetric Group” under my supervision and guidance.

This project has been undertaken as a part of the undergraduate CBCS (New) curriculum of

Mathematics (Honours), Semester: VI, Paper: DSE – 4, Course Title: Dissertation of any topic

in Mathematics (Project Work), Course ID: 62127and for the partial fulfillment of the degree of

Bachelor of Science (Honours) in Mathematics of Bankura University under CBCS (New)

Curriculum in 2024–25.

Signature of the Supervisor Signature of the HOD

Name: Department of Mathematics
Designation: Bankura Christian College

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 Declaration

I hereby declare that my project, titled “Symmetric Group”, submitted by me to Bankura

University for the purpose of DSE – 4 paper, in semester VI under the guidance of my
professor of Mathematics in Bankura Christian college, Dr. Arup Mukhopdhyay.

I also declare that the project has not been submitted hereby any other student.

Name: Bristi Patra

UID Number: 22013121019

Semester: VI

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 ACKNOWLEDGEMENT

First of all I am immensely indebted to my intention, Bankura Christian college

under Bankura University and it is my humble pleasure to acknowledgement my
deep senses of gratitude to our principal sir, Dr. Fatik Baran Mandal, for the
valuable suggestions and encouragement that made this project successful.

I am grateful to the HOD of our Mathematics department, Dr. Utpal Kumar Samanta
for always lending his helping hand in every situation. A heartful thanks to my guide
in this project, Dr. Arup Mukhopadhyay, whose guidance and valuable support has
been instrumental in the completion of this project work.

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 TABLE OF CONTENTS

SL.NO TOPICS PAGENO

1. INTRODUCTION 7-9

2. STRUCTURE OF Sn 10-11

3. CYCLENOTATION 12-13

4. TRANSPOSITION 13-14

5. SIMPLE TRANSPOSITION 14-15

6. ORDER OF ELEMENTS OF Sn 15-16

7. CONJUGACY CLASSES OF Sn 16-17

8. CLASS EQUATION OF Sn 18-19

9. IMPORTANT THEOREMS 20-24

10. SUB GROUPS OF Sn 26-27

11. OBJECTIVE AND 28-29

IMPORTANCE

12 CONCLUSION 30

13. REFERENCES 31

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 INTRODUCTION

❖ In the study of abstract algebra, group serves as one of the most fundamental structures for understanding
symmetry and mathematical operations. It consists of a set of element combined with an operation that
satisfies four basic properties: closure, associativity, identity and invertibility. Among the various types
of groups, the symmetric group, denoted by Sn, plays a particularly important role.

❖ In the context of the symmetric group, the word "symmetric" refers to the rearrangements or permutations
of elements that preserve structure through re-ordering rather than geometric symmetry (asinshapes). The
term "symmetric" here comes from the idea that permuting objects doesn't change the nature of the set,
just the order — similar to how geometric objects may look the same after certain symmetrical
transformations (like rotation or reflection).

❖ The group captures all the possible symmetries of a set of distinct objects under permutation. In geometry,
symmetry refers to an object looking the same after a transformation. In algebra, the symmetric group
captures "symmetries" of a finite set by listing all the ways elements can be re-arranged without
duplication.

❖ The symmetric group is a natural and powerful example of a finite non-abelian group, making it essential
for introducing and exploring key group-theoretic concepts such as identity, inverses, orders of elements,
subgroups, cycles, and conjugacy. Every element in can be expressed as a product of disjoint cycles, and
the study of cycle structure offers a clear way to understand the behavior of permutations.

❖ HISTORY

The study of symmetric groups, which are fundamental in group theory, emerged in the 18th and 19th centuries.
Mathematicians like Lagrange and Cauchy initially explored permutations, laying the ground work for the concept
of symmetric groups. Later, Évariste Galois and Arthur Cayley significantly advanced the theory, with Cayley
defining abstract groups. The work of Camille Jordan further popularized the term "group" and its applications.

Here's a more detailed breakdown:

• Early Explorations (18th Century):
Lagrange and others began investigating permutations and their properties, unknowingly
foundation for symmetric groups.

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• Formalization and Abstraction (19th Century):
Galois and Cayley made crucial contributions by formalizing the concept of a group and its abstract
properties.

• Jordan's Influence:
Camille Jordan's work on group theory, particularly his use of the term "group," significantly
impacted the field's development and wider recognition.

• Applications in Geometry:
Felix Klein's Erlangen program in 1872 linked symmetry groups to geometry, demonstrating the
broad applicability of the theory.

• Modern Significance:
Symmetric groups continue to be a central concept in various mathematical fields, including
algebra, combinatorics, geometry, and computer science. They are also used to model card
shuffling and other permutation-based processes.

❖ APPLICATION
Symmetric groups have found applications in diverse fields, including:

Algebra: As a foundational example in group theory.

Combinatorics: Modeling card shuffling and other permutation-based problems.

Geometry: Representing symmetries of geometric objects.

Computer Science: In areas like coding theory and cryptography.

❖ Thus, the symmetric group is more than just a set of permutations—it is a gateway into the deeper world
of algebraic structures and mathematical reasoning. This project aims to explore the definition, properties,
structure, and significance of symmetric groups, along with their applications and theoretical implications.

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• What is Group: Formally, a group is a pair (G,∗), where G is a set and ∗ is a binary operation on G
such that:

1. Closure: For all 𝑎, b ∈ G, the result of 𝑎∗b is also in G

2. Associativity: For all 𝑎, b, c ∈G, (𝑎∗b)∗c=(𝑎∗b)∗𝑐.

3. Identity element: There exists an element e∈G such that 𝑎∗ e=e∗𝑎=𝑎 for all 𝑎∈G

4. Inverse element: For each a∈G, there exists an element 𝑎−1 such that 𝑎∗𝑎−1=𝑎−1∗𝑎=e

o One of the richest sources of group theory insights comes from considering Permutation
Groups.

• What is Permutation Group: X be a nonempty Set. A permutation f of X is a one one

and onto map from X onto X.Agroup (G, ∗) is said to be a permutation group on a non empty
Set X if the elements of G are permutations of X and the operation ∗ is the composition of two
functions. The Symmetric group is one type of Permutation group. It is defined as the group of
all possible permutations (bijective functions) of a set with finite elements, under the
composition of function.

o Symmetric group (Sn) is a famous example of permutation group

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 THE STRUCTURE OF Sn

As always, S(n) is the group of bijections or permutations of a set of n objects, sayXn= {1, 2,…, n}. Its group
operation is the composition of bijections. We will frequently refer to the objects being permuted as letters. This
will be convenient for when we take up cryptography
Recall the notation

1 2 … 𝑛
𝑓=[ ]
𝑓(1) 𝑓(2) … 𝑓(𝑛)

To simplify the notation, we will notation by also write f=[f(1),f(2),…., f(n)]

❖ Composition of Permutation: Composition of permutations expressed in array notation is carried

out from right to left by going from top to bottom, then again from top to bottom.
For example, let f and g be two permutations on the set{1,2,3,4,5} by specifying f (1)=2, f (2)=4,
f(3) = 3, f(4) = 5, f(5) = 1

and

g(1)=5, g(2)=4, g(3)=1, g(4)=2, g(5)= 3

1 2 3 4 5 1 2 3 4 5
𝑡ℎ𝑎𝑡 𝑖𝑠, 𝑓 = ( ) 𝑎𝑛𝑑 𝑔 = ( )
2 4 3 5 1 5 4 1 2 3

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
𝑔𝑓 = ( )( )=( )
5 4 1 2 3 2 4 3 5 1 4 2 1 3 5

On the right we have 4 under 1, since (gf)(1)= g (f(1))= g(2) =4, so gf sends 1 to 4. The remainder of the
bottom row gf is obtained in a similar fashion.

❖ Properties:
➢ If f and g be two permutations on Xn then f∗g (here “∗” is the composition of two function) is also
A permutation on Xn

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➢ Since the composition of mappings is associative, multiplication of permutation is associative.
The identity permutation of Xn is 𝑖=(
1 2 3 … 𝑛
)
1 2 3 … 𝑛

1 2 3 … 𝑛
➢ Let f be a permutation on Xn such that 𝑓 = ( ) , the inverse of f is defined by
𝑓(1) 𝑓(2) 𝑓(3) … 𝑓(𝑛)

𝑓(1) 𝑓(2) 𝑓(3) … 𝑓(𝑛)

𝑓 −1 = ( )
1 2 3 … 𝑛

➢ Multiplication is not commutative in general. For example, let

1 2 3 4 … 𝑛 1 2 3 4 … 𝑛
𝑓=( ) 𝑎𝑛𝑑 𝑔 = ( )
2 1 3 4 … 𝑛 3 2 1 4 … 𝑛
1 2 3 4 … 𝑛 1 2 3 4 … 𝑛
𝑡ℎ𝑒𝑛 𝑓𝑔 = ( ) 𝑎𝑛𝑑 𝑔𝑓 = ( )
3 1 2 4 … 𝑛 2 3 1 4 … 𝑛

So fg ≠ gf

➢ Order of Sn = n!

Therefore Sn is a finite non abelian group for n≥ 3

SYMMETRIC GROUP S3:

S3be the set of all permutation on the set {1,2,3}. The elements of S3 are
ρ0 = 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
( ) , ρ1 =( ) , ρ2 =( ), ρ3 = ( () , ρ = ),
1 2 3 2 3 1 3 1 2 1 3 2 3 42 1

1 2 3
ρ5 = ( )
2 1 3

The multiplication table given below 𝜌0 𝜌1 𝜌2 𝜌3 𝜌4 𝜌5

𝜌0 𝜌0 𝜌1 𝜌2 𝜌3 𝜌4 𝜌5
𝜌1 𝜌1 𝜌2 𝜌0 𝜌5 𝜌3 𝜌4
𝜌2 𝜌2 𝜌0 𝜌1 𝜌4 𝜌5 𝜌3
𝜌3 𝜌3 𝜌4 𝜌5 𝜌0 𝜌1 𝜌2
𝜌4 𝜌4 𝜌5 𝜌3 𝜌2 𝜌0 𝜌1
𝜌5 𝜌5 𝜌3 𝜌4 𝜌1 𝜌2 𝜌0

It appears from the table that 𝜌0 is the identity element

The inverse of 𝜌0, 𝜌1, 𝜌2, 𝜌3, 𝜌4, 𝜌5 is 𝜌0, 𝜌2, 𝜌1, 𝜌3, 𝜌4, 5 respectively.
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 CYCLE NOTATION

There is another notation commonly used to specify permutations. It is called cycle notation and was first
introduced by the great French mathematician Cauchy in1815.Cycle notation has theoretical advantages in that
certain important properties of the permutation can be readily determined when cycle notation is used.

As an illustration of cycle notation, let us consider the permutation α of S6

1 2 3 4 5 6
𝛼=( )
2 1 4 6 5 3

This assignment of values could be presented schematically as follows.

Although mathematically satisfactory, such diagrams are cumber some. Instead, we leave out the arrows and simply write
α = (1, 2)(3,4,6) (5) . As a second example, consider

1 2 3 4 5 6
𝛽=( )
5 3 1 6 2 4

In cycle notation, β can be written (2,3,1,5) (6,4) or (4,6) (3,1,5,2), since both of these unambiguously
Specify the function β.

An expression of the form (a1, a2,.. .,am) is called a cycle of length m or an m-cycle.

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 ➢ Cycle Decomposition: Every permutation in S(n) can be written as a product (composition)
of disjoint cycles. Disjoint cycles operate on mutually exclusive sets of elements and hence
commute with each other. For example, the permutation

𝜎 ∈ S6 given by:

𝜎=(145)(26)
1 4, 4 5 ,5 1,2 6,6 2,3 3

➢ Properties of Cycle Notation:

o Uniqueness (up to order and rotation) The decomposition of a permutation into disjoint cycles
is unique up to the order in which the cycles are written and cyclic rotations within each cycle.
For instance, (3 5 1), (5 1 3) and (1 3 5), and all represent the same cycle.
o Disjoint Cycles Commute If two cycles are disjoint (i.e.,no elements are shared), then they
commute:
(a1a2…..ak) (b1b2…..bm) = (b1b2…..bm) (a1a2…..ak)

o Identity Permutation The identity permutation, which fixes every element, is often denoted
by an empty cycle or ( ) , and it acts trivially on all elements.
o Inverses in Cycle Notation The inverse of a cycle is obtained by reversing the order of its
elements. That is

(a1a2…..ak)-1= (ak-1ak…. a1)

TRANSPOSITION

We now introduce a set of building blocks for the symmetric group. These are called transpositions.

A permutation which interchanges two letters i and j and leaves all the other letters unchanged is called a
transposition. The transposition which interchanges i and j will be denoted by (ij).

Clearly, transpositions are the simplest permutations since if f ∈Sn moves the letter i to j, then j also is permuted
to something else. Notice that by our convention that Sn⊂Sn+1, the transposition (ij) ∈ Sn , for all n such that i, j
≤n. Transpositions are easy to multiply. For example, (23)(12) represents the permutation

1 2 3,2 1 1,3 3 2

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 Hence, (23) (12) = (312)

Obviously there are two ways to write a transposition: (ij) = (ji): Also, (ij)-1= (ji): each transposition is its own
inverse.

Example: Consider S3. We saw above that (23)(12)=(312). Taking the product in the other order gives a different
result, namely (12) (23) = (2 3 1)

Hence (23) (12) ≠ (12) (23)

o Every element of Sn can be represented as a product of permutation

For example: consider the permutation f in S5 where f can be written as

f = (1 23 4 5) = (15) (14) (13) (12)

i.e. in general in Sn, (a1a2……am) = (a1am) (a1am-1)…..(a1a2)

SIMPLE TRANSPOSITION

Certain transpositions can be further decomposed into transpositions. For example, (13) = (12) (23) (12). The
special transpositions used here, i.e. those of the form
(ii+1), are called simple transpositions.
For another example,
(14) = (12) (23) (34) (23) (12).

These are particularly important because any permutation in Sn can be written as a product of simple

transposition For example: let Let’s consider the permutation:

𝜎 = (1 4 3)
We will express this as a product of simple transpositions, i.e., transpositions like (12), (23) and (34)etc.

𝜎 = (1 43) = (13) (14)

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 Now we express:

(13) = (23) (12) (23)

(14) = (34) (23) (12) (23) (34)

Hence

(14 3) = (23) (12) (23) (34) (23) (12) (23) (34)

ORDER OF ELEMENTS OF Sn

Order of Element: In group theory, the order of an element α is the smallest positive integer k such that αk=
identity

The order of a permutation in Sn is the least common multiple (LCM) of the lengths of the disjoint cycles in its
composition

Example 1: Single Cycle

𝜎 = (1 23 4) ∈ S4

Length of 𝜎 is 4, order = 4

Example 2: Disjoint Cycle

α = (1 34) (25) ∈ S5
order = LCM (3,2)=6

Example 3: Transposition

α = (1 3) ∈ S2
order = 2

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 • General Formula for counting Cycle type:

If a permutation has Cycle Type

m1 cycles of length 1

m2 cycles of length 2

m3 cycles of length 3

.....

Mk cycles of length k
such that:

1.m1 + 2.m2 +…. + k.mk = n

Then the number of such permutation in Sn is:

𝑛!
{𝑚1} {𝑚2} {𝑚 }
(1 1!·2 𝑚2!·…·𝑘 𝑘 𝑚!) 𝑘

For example in: the number of permutation( e) (ab) (cd) i.e., two 2 cycle and one 1 cycle

5! 120
Then: = =15
(111!·22 2!) 1.1.4.2

CONJUGACY CLASSES OF Sn

o Definition of conjugacy class:

Let G be an arbitrary group. Two elements a, b of G are said to be G-conjugate if there is an x∈G such
that b = xax-1. The set of all elements of G conjugate to a is called the G-conjugacy class of a, or
simplythe conjugacy class of a , denoted by cl(a).

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 o Conjugacy in Sn:

➢ Let and suppose 𝜎 𝑎𝑛𝑑 𝜋 ∈ Sn has cycle decomposition (a1a2….ak1) (b1b2….bk2)…..,

Then 𝜋𝜎𝜋-1 has the cycle decomposition (𝜋 (a1) 𝜋 (a2)…. 𝜋 (ak1)) (𝜋 (b1) 𝜋 (b2)…. 𝜋 (bk2))...

➢ If 𝜎∈Sn is the product of disjoint cycles of lengths n1,n2,….,nr with n1≤n2≤n3….≤nr

(including its 1-cycles),
Then the integers n1,n2,….,nr are called the cycle type of 𝜎

For example let 𝜎 = (1) (23) (4 56) ∈ Sn Then the cycle type of is 1,2,3

➢ If n is a natural number, a partition of n is any increasing sequence of positive integers whose

sum is n.

➢ Two elements of Sn are conjugate in Sn iff the y have the same cycle type. The number
of distinct conjugacy classes of Sn equals the number of partition of n

EXAMPLE: Conjugacy classes and there representative in S5

PARTITION OF 5 REPRESENTATIVE OF DISTINCT CONJUGACY

CLASS
1+1+1+1+1 e
1+1+1+2 (12)
1+2+2 (1 2)(34)
1+1+3 (1 23)
1+4 (1 2 34)
2+3 (12)(3 45)
5 (1 2 34 5)

WHAT IS CLASS EQUATION OF A GROUP G

Let G be a finite Group and a ∈G. Z(G) be it’s centre where Z(G)={ x∈ G:xg=gx, for all g ∈G}

So the class equation of G is

|G|=|Z(G)|+∑𝑎∈𝐴−𝑍(𝐺)|𝑐𝑙(𝑎)|
Where A is the subset of G which contains exactly one element from each conjugacy class

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 ➢ Centre of Sn only contains the identity element i.e. Z(Sn) = {e}

CLASS EQUATION OF Sn
Class equation of Sn is

|Sn|=|Z(Sn)|+∑𝑎∈𝐴−𝑍(Sn)|𝑐𝑙(𝑎)|
Where A is the subset of G which contains exactly one element from each conjugacy class

EXAMPLES:

• CLASS EQUATION OFS4

PARTITION OF 4 REPRESENTATIVES OF NUMBER OF ELEMENTS IN

DISTINCT CONJUGACY THE CLASS
CLASSES
1,1,1,1 e 1

1,1,2 (12) 4!
=6
(122!·211!)
1,3 (1 23) 4!
=8
(111!·311!)
2,2 (1 2)(34) 4!
=3
(222!)
4 (1 2 34) 4!
1 =6
(4 1!·)

Z (S4) = {e} order of S4 = 4! =24

The distinct conjugacy classes of S4 is cl{e}, cl{(1 2)}, cl{(1 23)}, cl{(12) (34)}, cl{(123 4)}

Therefore the class equation of S4 is

24 = 1 +6 +8+3+6
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 • CLASS EQUATION OF S5

PARTITIONOFS5 REPRESENTATIVES OF NUMBER OF ELEMENTS IN

DISTINCT CONJUGACY THE CLASS
CLASSES
1,1,1,1,1 (e) 1

1,1,1,2 (12) 10

1,2,2 (1 2) (34) 15

2,3 (12) (3 45) 20

1,1,3 (1 23) 20

1,4 (1 2 34) 30

5 (1 2 34 5) 24

Order of S5 is 5! = 120

Z(S5) = {e}

Therefore, the class equation of S5is

120 =1 +10 +15 +20+20 +30 +24

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 IMPORTANT THEOREMS

Theorem1:
Order of Sn is n!
Proof.

The symmetric group Sn is the group of all permutations of a set, of n elements, typically {1,2,3,…,n}.
A permutation is a bijective function from the set to itself.

To prove that |Sn| = n!

We count how many different permutations (bijective functions) of n elements exist.

Let’s construct a permutation step-by-step:

• There are n choices for the image of 1(where1 gets mapped to).

• After choosing the image of 1,there are n−1 remaining choices for the image of 2.
• Then n−2 choices for the image of 3.

• ...
Finally, only 1 choice for the image of n. So, the total number of different permutations is

n⋅(n−1)⋅(n−2)⋯1 = n!

Since each element of Sn corresponds to exactly one permutation of n distinct elements, and the number of such
permutations is n!, we conclude:

∣Sn∣ = n!

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Theorem 2:
Any non identity permutation π of Sₙ(n≥2) can be uniquely expressed(up to the order of
the factors) as a product of disjoint cycles, where each cycle is of length at least 2.

Proof.
We prove the result by induction on n. Suppose n=2. Now |S₂|=2 and then on identity
Element of S₂ is
1 2
𝛼=( ) Now α = (1 2), i.e., α is a cycle.Thus, the theorem is true for n = 2.
2 1
Suppose n>2 and the theorem is true for all Sₖ such that 2≤k<n. Let π be a non identity
Element of Sₙ. Now πⁱ(1)∈ Iₙ for all integers i, i≥1. Therefore, {π(1),π²(1),...,πⁱ(1),...}
⊆Iₙ. Since Iₙ is a finite set, we must have πl(1)= πᵐ(1) for some integers l and m such
That l>m≥1.This implies that πˡ⁻ᵐ(1) = 1. Let us write j = l-m.Then j>0 and πʲ(1)=
1. Let i be the smallest positive integer such that πⁱ(1)=1.Let

A={1, π(1),π²(1),..., πⁱ⁻¹(1)}.

Then all elements of the set A are distinct. Let τ∈Sₙ be the permutation defined by

τ=(1 π(1)π²(1) ...πⁱ⁻¹(1)),

i.e., τ is a cycle. Let B = Iₙ \ A. If B = ∅, then π is a cycle. Suppose B ≠ ∅. Let σ = π

restricted to B. If σ is the identity, then π is a cycle. Suppose that σ is not the identity.
Now by the induction hypothesis, σ is a product of disjoint cycles on B, say,

σ = σ₁ ∘σ₂ ∘... ∘σᵣ.

Now for 1≤i≤r, define πᵢ by

πᵢ(a) = σᵢ(a) if a ∈ B,
a if a ∉ B

Then π₁,π₂,...,πᵣ and τ are disjoint cycles in Sₙ. It is easy to see that

π=π₁ ∘π₂∘... ∘πᵣ ∘τ.

Thus, π is a product of disjoint cycles.

To prove the uniqueness, let

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π=π₁ ∘π₂∘... ∘πᵣ =μ₁∘μ₂ ∘... ∘μₛ,

a product of r disjoint cycles and also a product of s disjoint cycles, respectively.We

show that every πᵢ is equal to some μⱼ and every μₖ is equal to some πₗ.

Consider πᵢ,1≤i≤r. Suppose πᵢ

= (i₁ i₂ ... iₗ).

Then πᵢ(i₁)≠i₁. This implies that i₁ is moved by some μⱼ. By the disjointness of the cycles,
there exists unique μⱼ, 1 ≤ j ≤ s, such that i₁ appears as an element in μⱼ. By reordering, if
necessary, we may write

μⱼ = (i₁c₂ ... cₘ).

Now

i₂ = πᵢ(i₁) = π(i₁) = μⱼ(i₁) =c₂

i₃ = πᵢ(i₂) = π(i₂) = π(c₂) = μⱼ(c₂) =c₃
...
iₗ = πᵢ(iₗ₋₁) = π(iₗ₋₁) = π(cₗ₋₁) = μⱼ(cₗ₋₁) =cₗ

If l<m, then i₁ = πᵢ(iₗ) = π(iₗ) = π(cₗ) = μⱼ(cₗ) =cₗ₊₁, a contradiction.

Thus, l=m. Hence, πᵢ=μⱼ for some j, 1≤j≤s. Similarly, every μₖ=πₜ for some t,1≤t≤
r.

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Theorem 3:
If n>1, every element of Sn can be represented in some (non-unique) way as a product of
transpositions.

Proof. Let σ ∈ S(n) be represented as σ = [σ(1),σ(2),...,σ(n)]. Suppose σ(n) = k. Then

(kn) σ(n) = n, and we have

(kn) σ = [σ(1),σ(2),...,σ(n),...,σ(n−1),n],

Where σ(n) is the k-th component above. Now let us induct on n. The result is clear if n=
2, so let’s suppose it’s true for n − 1 where n ≥ 3. But then with σ as above, σ′ = [σ(1),
σ(2),...,σ(n−1)] ∈ Sn-1 can be represented as a product of transpositions lying in Sn-1, say
σ′=t₁⋯tₘ. Hence, σ = (kn)t₁⋯tₘ is a representation of σ as a product of transpositions.

This completes the proof.

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Theorem 4:

Cayley's Theorem
Statement:

Every group G is isomorphic to a subgroup of the symmetric group acting on G.That is,
for any group G, there exists a subgroup H of the symmetric group SG such that G≅ H.

Proof:

Let G be any group. Define a function φ:G→Sym(G), where Sym(G) is the symmetric
group on the set G. For each element g ∈ G, define φ(g): G → G by left multiplication:

φ(g)(x) =g * x for all x ∈G

We will show that φ is a group homomorphism from G to Sym(G), and that φ is injective.
Hence, the image of φ is a subgroup of Sym(G) isomorphic to G.

Step1: φ(g)is a bijection

For any g∈G, the map φ(g)(x) = g*x is a bijection because every group operation has
An inverse. The inverse map is φ(g⁻¹) (x) = g⁻¹*x.

Step 2: φ is a homomorphism

For g₁, g₂∈G, and for any x∈G:

φ(g₁g₂) (x) = (g₁g₂)*x = g₁*(g₂*x) = φ(g₁)(φ(g₂)(x)) So,
φ(g₁g₂) = φ(g₁) ∘ φ(g₂)
Hence, φ is a homomorphism.

Step3: φ is injective

Suppose φ(g) = φ(h). Then for all x∈ G, φ(g)(x) = φ(h)(x), i.e., g*x=h*x. In
particular, let x = e (the identity in G):
g*e = h*e ⇒ g=h Hence,
φ is injective.

Conclusion:

Since φ is an injective homomorphism from G to Sym(G), the image of G under φ is a

subgroup of Sym(G) that is isomorphic to G. Therefore, every group is isomorphic to a
subgroup of a symmetric group.

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 ▪ Even and Odd Permutations
In the symmetric group Sₙ, every permutation can be classified as either an even
permutation or an odd permutation.

A transposition is a permutation that swaps two elements and leaves all others fixed.Any
permutation in Sₙ can be expressed as a product of transpositions.

A permutation is said to be even if it can be written as a product of an even number of

transpositions.

A permutation is said to be odd if it can be written as a product of an odd number of

transpositions.

The parity (evenness or oddness) of a permutation is well-defined, meaning that if a

permutation can be expressed using both an even and an odd number of transpositions,
then the permutation is the identity. Otherwise, the number of transpositions used in any
decomposition must be either always even or always odd.

Properties:

1. The identity permutation is even (it is the product of zero transpositions).

2. The product of two even permutations is even.

3. The product of two odd permutations is even.

4. The product of an even and an odd permutation is odd.

5. Inverse of an even permutation is even and inverse of a odd permutation is odd

6. The number of even permutation on a finite set (contains at least two elements) is
equal to the number of odd permutation on it.

Example:

Let π = (123). This 3-cycle can be written as a product of two transpositions: (13) (12).
Hence, π is an even permutation.

Let σ = (12). This is a single transposition and hence an odd permutation.

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 SUBGROUPS OF Sn

▪ Definition of Subgroup
Let (G,∗) be a group and H be an on empty subset of G. If (H,∗) is a group where, ∗ is
the composition of function , then ( H ,∗) be a subgroup of (G,∗)

▪ Subgroup generated by an element

Let G be a group and H be a subgroup of G, let a∈H, Then H is said to be generated by
a if H = { an: n ∈ Z } i.e. H = <a>

SUBGROUP LATTICE OF S3
S3={e, (12), (23) ,(13), (1 23), (1 32)}

The subgroup lattice of S3

Where,

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 <f1> = <(12)> = {e, (12)}

<f2> = <(23)> = {e, (23)}

<f3> = <(13)> ={e, (13)}

<r1> = <(1 23) > ={ e,(1 2 3) ,(1 3 2) } = <(1 3 2)>

❖ NORMAL SUBGROUP OF Sn

Definition of Normal Subgroup: A subgroup H of a group G is said to be

normal subgroup of G if aH = Ha holds for all a ∈ G.

Proposition: let H be a subgroup of G and [G:H]=2, then H is normal in G.

• ALTERNATING GROUP (An)

Set off all even permutation of Sn forms a Group with respect to composition of
permutation. This group is called Alternating Group of degree n and it is denoted by An

➢ Let f and g be two even permutation of An then fg is also an even permutation

i.e. fg ∈An.
➢ Since composition of permutation is associative, composition is associative
in An.
➢ The identity permutation is an even permutation.
➢ Inverse of an even permutation is even so iff ∈An, f-1∈An.
➢ Thus An forms a Group with respect to composition of permutation, Hence
An is a subgroup of Sn .
➢ Since composition of permutation is not commutative in general, An is a non
Abelian group for n≥ 4.
In Sn the number of even permutation is equal to the number of odd permutation. Therefore, the number of
𝑛!
even permutation in Sn= 2
𝑛!
Hence, |An|= 2

|𝑆𝑛|
An is a subgroup of Sn and [Sn:An] = |𝐴𝑛|
Hence An is a normal subgroup of Sn

NOTE: The set of all odd permutations of Sn does not form a Group under composition
of permutation because It is not closed under multiplication also it does not contain
identity element.

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 OBJECTIVE AND IMPORTANCE
Objective of the Project
The primary objective of this project is to explore and analyze the concept of symmetric
groups, which play a central role in modern algebra, especially in the study of group
theory and permutations. Symmetric groups, denoted by Sn, consist of all possible
permutations of n distinct elements and are foundational in understanding the structural
properties of various algebraic systems.

This project aims to:

1. Understand the fundamental concepts of permutations and their operations.

2. Define and investigate symmetric groups Sn in terms of elements, operations,

orders, and generators.

3. Explore subgroup structures of symmetric groups, including cyclic

subgroups, alternating groups, and normal subgroups.

4. Analyze the cycle structure of permutations, including disjoint

cycle decompositions and their impact on the order of elements.

5. Classify permutations into even and odd types and understand the concept
of parity and the alternating group.

6. Study the applications of symmetric groups in solving problems in abstract

algebra, combinatorics, and even real-world contexts such as cryptography
and chemistry.

7. Develop a comprehensive understanding of the algebraic structure and

significance of symmetric groups through examples, theorems, and
graphical representations such as Cayley tables and subgroup lattices.

Through this project, the objective is not only to understand the symmetric group in a
theoretical context but also to appreciate its role in broader mathematical frameworks
and real-world applications.

Importance of the Study

The study of symmetric groups is of great significance in abstract algebra and
mathematics as a whole. Their importance can be highlighted in the following ways:

1. Foundation of GroupTheory: Symmetric groups are among the first and most
important examples of finite groups. Every finite group is isomorphic to a

28
 Subgroup of some symmetric group (Cayley’sTheorem), making symmetric
groups fundamental to the theory.

2. Understanding Permutations: Permutations are key in various mathematical

processes, such as combinatorics, probability, and algebra. Symmetric groups
provide a structured way to study permutations and their algebraic behavior.

3. Structure and Classification: Symmetric groups help in classifying group

elements based on cycle types and parity. This leads to deeper insights into
the order and structure of algebraic systems.

4. Applications in Mathematics and Science:

o In combinatorics, symmetric groups are used to count

distinct arrangements.

o In Galois theory, symmetric groups are linked to solvability of

polynomial equations.

o In chemistry and physics, they are used to model molecular symmetries.

o In computer science, permutation groups play roles in sorting

algorithms and cryptography.

5. Theoretical Development: Symmetric groups provide examples and counter

examples that are crucial in the development of modern algebraic concepts like
normal subgroups, homomorphisms, isomorphisms, and quotient groups.

6. Gateway to Advanced Topics: A deep understanding of symmetric groups

opens the door to more advanced mathematical topics such as representation
theory, group actions, and algebraic topology.

In summary, studying symmetric groups is not only academically enriching but also
essential for pursuing higher studies in mathematics, theoretical physics, and computer
science. This project aims to bridge the gap between abstract theory and practical
applications, illustrating the power and elegance of symmetry in mathematical structure.

29
 CONCLUSION
The study of symmetric groups offers a deep and fascinating insight into the structure
and behavior of mathematical systems governed by permutations. Throughout this
project, we have explored the foundational concepts of symmetric groups SnS_nSn,
including their elements, operations, subgroup structures, and the classification of
permutations into even and odd types. We have seen how every symmetric group
encapsulates all the possible rearrangements of a finite set, making it a central objectof
study in group theory.

By analyzing the properties and internal structure of symmetric groups, we developed an

understanding of important concepts such as cycle notation, disjoint cycles, order of
elements, alternating groups, and simple transpositions. These tools not only help us
work within symmetric groups but also extend to more advanced areas of mathematics.

One of the most significant outcomes of this project is recognizing the universal
importance of symmetric groups. They are not just theoretical constructs but serve as
powerful models in various fields—ranging from combinatorics and algebra to physics,
computer science, and chemistry. The presence of symmetry is a natural phenomenon,
and symmetric groups provide a formal language to study and express this symmetry in a
rigorous mathematical framework.

In conclusion, symmetric groups are a cornerstone of abstract algebra. They provide a

rich ground for both theoretical exploration and practical application. Understanding
symmetric groups not only enhances our knowledge of group theory but also strengthens
our mathematical reasoning and problem-solving abilities, making it an essential topic
for further studies and research in pure and applied mathematics.

30
 REFERENCES
• Gallian, JosephA. Contemporary Abstract Algebra. Cengage Learning.
• Dummit, DavidS., and Richard M.Foote. Abstract Algebra.
• Artin, Michael. Algebra. Pearson Education.
• Online resources such as Khan Academy, Brilliant. org, and Math World.

• Wikipedia: Symmetric Group

https://en.wikipedia.org/wiki/Symmetric_group

• MIT Open Course Ware –Abstract Algebra

https://ocw.mit.edu

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