Page 67 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 5: Normal Subgroups

CONTENTS
Objectives

Introduction
5.1 Normal Subgroups
5.2 Quotient Groups
5.3 Summary
5.4 Keywords

5.5 Review Questions
5.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss the concept of normal subgroups

Explain the Quotient group

Introduction

In earlier units, you have studied about the term subgroups and cosets. In this unit, we will
discuss a special class of subgroups known as normal subgroups. You will also come to know
about that the cosets of such a subgroup form a group with respect to a suitably defined operation.
These groups are called quotient groups. After discussing these concepts, we will also discuss
some examples related to this concept.

5.1 Normal Subgroups

In the last unit, you have studied about coset of a subgroup also introduced with a fact that left
coset aH, not be same as the right coset Ha.
But this fact is true for certain subgroup for which Ha and aH represented by the same element
coincide.
In group theory, these types of subgroup are very important and this type of a subgroup has a
special name. This subgroup is referred to normal subgroup.
Definition: A subgroup N of a group G is called a normal subgroup of G if Nx = xN  x  G, and
we write this as N  G.
For example, any group G has two normal subgroups, namely, {e} and G itself. Can you see
why? Well, {e}x = {x} = x{e}, for any x  G, and Gx = G = xG, for any x  G.

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