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Permutation groups are groups whose elements are permutations of a given set, and the group operation is composition of permutations, permutation groups arise in many different areas of mathematics, including combinatorics, graph theory, and representation theory.
This project basically gives a brief account of Groups with particular emphasis on permutation groups, and its properties. This project covers some basic applications of permutation groups in other fields.

Table of Contents 
Table of Contents vii

1.1 Background of the Study 1
1.2 Statement of the Problem 3
1.3 Significance of the Study 4
1.4 Aims and Objectives of the Study 5
1.5 Scope and Limitations 6
1.6 Definitions of Basic Terms 7


3.1 Group 13
3.2 Permutation 14
3.3 Symmetric Group 15
3.3.1 conjugacy classes 15
3.4 Alternating Group 16
3.4 Even and Odd Permutation 17
3.5 Cycles 17
3.6 Transposition 18
3.7 Disjoint Cycles 18
3.7.1 Product of Disjoint Cycle 18
3.8 cyclic group 19
3.9 Some properties of permutation group 19

4.1 The Symmetry 21
4.1.1 Permutation of an Equivalent Triangle 21
4.1.2 Permutation of A Square 26
4.1.3 Permutation of Hexagon 31
4.2 Other Applications of permutation groups 36
4.2.1 Latin-Square 36
4.2.1 Arrangement 36

Summary, Conclusion and Recommendations
5.1 Summary 38
5.2 Conclusion 38
5.3 Recommendation 39


1.1 Background of the Study 
In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, and may be called the symmetric group on n letters.

By Cayley's theorem, every group is isomorphic to some permutation group.

The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.

Permutation groups have a rich history in mathematics, dating back to ancient times when they were used to solve combinatorial problems. However, their formal study gained prominence during the development of group theory in the 19th century. Mathematicians like Augustin Louis Cauchy and Évariste Galois made significant contributions to understanding permutation groups as symmetries of mathematical objects.

The exploration of permutation groups and their algebraic properties has far-reaching applications beyond mathematics itself. In computer science, permutation groups are used in cryptography, sorting algorithms, and designing efficient data structures. Additionally, they play a fundamental role in the study of molecular symmetries in chemistry and the analysis of symmetry in physics and crystallography.

Permutation groups are a fundamental object in abstract algebra, and their algebraic properties have been studied by mathematicians for centuries.

One of the earliest works on the algebraic properties of permutation groups was done by Leonhard Euler in the 18th century. Euler studied the symmetry of polyhedra and other geometric objects using permutation groups. He also developed a number of theorems on the properties of permutation groups, such as the Cayley-Hamilton theorem and the Jordan-Hölder theorem.

In the 19th century, Augustin-Louis Cauchy and Camille Jordan made significant contributions to the study of permutation groups. Cauchy developed a theory of groups that was based on the concept of a permutation group. Jordan developed a classification of finite permutation groups up to isomorphism.
In the 20th century, there was a great deal of progress made in the study of permutation groups. Some of the most important figures in this area include William Burnside, Alfred H. Clifford, and Richard Brauer.

Burnside classified all finite permutation groups of prime degree. Clifford and Brauer developed a theory of representations of permutation groups, which has many applications in other areas of mathematics.

Other important figures in the study of permutation groups in the 20th century include Walter Feit, Marshall Hall, and John Thompson. Feit and Thompson proved the famous Feit-Thompson theorem, which states that every finite non-abelian simple group contains an element of prime order.

The study of groups originally grew out of an understanding of permutation groups. Permutations had themselves been intensively studied by Lagrange in 1770 in his work on the algebraic solutions of polynomial equations. This subject flourished and by the mid 19th century a well-developed theory of permutation groups existed, codified by Camille Jordan in his book Traité des Substitutions et des Équations Algébriques of 1870. Jordan's book was, in turn, based on the papers that were left by Évariste Galois in 1832.

1.2 Statement of the Problem 
Problem statement: While the algebraic structures of permutation groups have been extensively studied, there is a gap in our understanding of how specific types of algebraic structures manifest within permutation groups and how they relate to the underlying symmetries and transformations. This study aims to investigate and analyze the algebraic structures that emerge from permutation groups and their implications in diverse mathematical contexts.

To tackle this problem, the study will focus on identifying and characterizing algebraic properties within permutation groups, such as the existence of subgroups with certain properties, cyclic structures, and relationships between permutation group operations and algebraic operations. Furthermore, the study will delve into the connections between these algebraic structures and various applications, such as cryptography, group representations, and combinatorics.

By addressing this problem, the research aims to contribute to the broader understanding of permutation groups as algebraic entities and uncover potential applications of these structures in both theoretical and practical domains. This investigation holds the potential to bridge the gap between algebraic structures and permutation groups, enhancing our knowledge of symmetries and transformations in mathematics and beyond.

1.3 Significance of the Study
Advancement of Mathematical Understanding: Investigating the algebraic properties of permutation groups can provide insights into the underlying symmetries and transformations of mathematical objects. By uncovering new relationships and properties, this study has the potential to enrich the field of algebra, contributing to group theory and abstract algebra. The results could lead to the development of new mathematical concepts and theorems, expanding the frontiers of mathematical knowledge.

Bridge between Theoretical and Applied Mathematics: Algebraic structures are not only abstract mathematical constructs but also have practical implications. By understanding how these structures emerge from permutation groups, this research could bridge the gap between theoretical and applied mathematics. The findings might lead to improved cryptographic algorithms, more efficient data processing techniques, and innovative solutions to combinatorial and optimization problems.

Impact on Computational Sciences: Algebraic properties within permutation groups have direct applications in computer science and computational mathematics. This study's insights could contribute to the design of algorithms for tasks like sorting, searching, and graph analysis. Moreover, understanding the algebraic aspects of permutation groups can aid in developing more robust and secure cryptographic systems.

Foundations for Further Research: As a foundational exploration, this study could inspire and guide further research endeavors. It might stimulate researchers to explore specific types of algebraic properties within permutation groups, analyze their behavior under different operations, and investigate their applications in specialized mathematical contexts.

Educational and Pedagogical Value: The study's outcomes could have educational implications, enriching curricula in abstract algebra, group theory, and related disciplines. The insights gained could serve as valuable teaching materials, demonstrating real-world applications of algebraic concepts and fostering a deeper understanding of mathematical structures.

In summary, the significance of this study lies in its potential to deepen our understanding of algebraic properties within permutation groups, paving the way for mathematical discoveries, practical applications, and educational enrichment. By exploring the connections between algebra and permutation groups.

1.4 Aim and Objectives of the Study
Characterize Algebraic Properties: The study aims to identify and characterize various algebraic properties within permutation groups. 

Examine Group Operations: Another objective is to analyze the interactions between the group operations of permutation groups and algebraic operations. This involves studying the effects of operations like composition and inversion on the algebraic properties and structures that emerge. 

Explore Applications: The study aims to connect algebraic structures within permutation groups to practical applications.

Establish Theoretical Connections: This objective involves establishing connections between the algebraic structures within permutation groups and other areas of mathematics. The research will explore relationships between permutation groups and concepts such as group representations, vector spaces, and module theory. By doing so, the study aims to contribute to the broader landscape of mathematical knowledge.

Contribute to Mathematical Literature: The research intends to contribute to the existing mathematical literature by presenting new findings, theorems, and insights related to algebraic properties within permutation groups.

In summary, the objectives of this research are multifaceted, encompassing the exploration of algebraic properties, the analysis of group operations, the investigation of applications, the establishment of connections, and the dissemination of knowledge. 

1.5 Scope and Limitations
1. The commutativity of finite permutation groups of small degree. This scope would limit you to studying finite permutation groups with a small number of elements.

The applications of permutation groups in a specific area of mathematics, such as combinatorics or graph theory. This scope would limit you to studying applications of permutation groups in a particular area of mathematics.

1.6 Definitions of Basic Terms
To ensure a clear understanding of the terminology used in this study, key concepts and terms are defined as follows:

1.6.1 Permutation Groups: 
A permutation group is a mathematical group that consists of all possible permutations of a set. In other words, it encompasses all possible arrangements or reorderings of the elements in the set. Permutation groups play a crucial role in group theory and have applications in combinatorics, cryptography, and other areas.

1.6.2 Algebraic properties: 
Algebraic properties are basic mathematical properties that apply to numbers, operations, and equations. They are used in many areas of mathematics, including algebra, calculus, and real analysis

1.6.3 Abstract Algebra:
Abstract algebra is a branch of mathematics that studies algebraic structures and their properties in a general and abstract manner. It deals with concepts that are applicable to various mathematical objects, allowing for the exploration of common underlying principles.

1.6.4 Group Operations: 
Group operations, such as composition and inversion, define how elements within a group interact with each other. These operations satisfy specific properties, including closure, associativity, identity, and inverse existence.

1.6.5 Cyclic Subgroups: 
A cyclic subgroup is a subgroup of a group that is generated by a single element. It consists of all powers of that element and forms a subset of the larger group.

1.6.6 Set: 
A set is defined to be a collection of well-defined distinct elements. Sets are usually denoted by capital letters like A, B, C, etc. while members of the sets are denoted by small letters a, b, c, etc. E.g.
Set of integers Z={.  .  .  .-2,-1,0,1,2,.   .   .  .}

1.6.7 Order: 
This is defined to be the number of elements in a set.
E.g. the set S={a,b,c} is of order 3.

1.6.8 proper subgroup:
A proper subgroup of a group G is a subgroup H of G such that H=G. In other words, a proper subgroup is a subset of G that satisfies all the group axioms, but it is not equal to the whole group. 

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