Page 97 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Richa Nandra, Lovely Professional University

Notes Unit 8: Permutation Groups

CONTENTS
Objectives

Introduction
8.1 Symmetric Group
8.2 Cyclic Decomposition
8.3 Alternating Group
8.4 Cayley’s Theorem

8.5 Summary
8.6 Keywords
8.7 Review Questions

8.8 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept of permutation group

Explain the symmetric group

Describe the cyclic decomposition

Prove and use Cayley’s Theorem

Introduction

In earlier classes, you have studied about the symmetric group. As you have often seen in
previous units, the symmetric groups S, as well as its subgroups, have provided us a lot of
examples. The symmetric groups and their subgroups are called permutation groups. It was the
study of permutation groups and groups of transformations that gave the foundation to group
theory. In this unit, we will prove a result by the mathematician Cayley, which says that every
group is isomorphic to permutations group. This result is what makes permutation groups so
important.

8.1 Symmetric Group

In earlier units, you have studied that a permutation on n non-empty set X is a bijective function
from X onto X. We denote the set of all permutations on X by S(X).
Suppose X is a finite set having n elements. For simplicity, we take these elements to be
1, 2, . . . , n. Then, we denote the set of all permutations on these n symbols by S .
n

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