A STUDY ON ALGEBRAIC PROPERTIES OF PERMUTATION GROUP

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Study Level: BTech, BSc, BEng, BA, HND, ND or NCE

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# ABSTRACT

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.

DECLARATION.. ii

CERTIFICATION.. iii

DEDICATION.. iv

ACKNOWLEDGEMENTS. v

ABSTRACT. vi

CHAPTER ONE. 1

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

CHAPTER TWO.. 10

LITERATURE REVIEW… 10

CHAPTER THREE. 13

ALGEBRAIC PROPERTIES AND PERMUTATION GROUPS. 13

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

CHAPTER FOUR.. 21

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

CHAPTER FIVE. 38

Summary, Conclusion and Recommendations. 38

5.1 Summary. 38

5.2 Conclusion. 38

5.3 Recommendation. 39

REFERENCES. 40

# INTRODUCTION

## 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 ÉquationsAlgébriques 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.

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