Page 77 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 6: Group Isomorphism

CONTENTS
Objectives

Introduction
6.1 Homomorphisms
6.2 Isomorphisms
6.3 Group Isomorphism
6.4 Summary

6.5 Keywords
6.6 Review Questions
6.7 Further Readings

Objectives

After studying this unit, you will be able to:
Explain the concept of homomorphism

Describe Isomorphism

Introduction

In the last unit, you have studied about the normal groups and the concept of quotient group.
In this unit, we will discuss various properties of those functions between groups which preserve
the algebraic structure of their domain groups. These functions are called group
Homomorphisms. This term was introduced by the mathematician Klein in 1893. This concept
is analogous to the concept of a vector space homomorphism, as you studied in the earlier unit.
In this unit, you will also get an idea about a very important mathematical idea—isomorphism.

6.1 Homomorphisms

Let us start our study of functions from one group to another with an example.
Consider the groups (Z, f) and ({1, - 1},). If we define

 1, if n is even
f : Z {1, –1} by f(n) =  1, if n is odd,
 
then you can see that f(a + b) = f(a).f(b)  a, b  Z. What we have just seen is an example of a
homomorphism, a function that preserves the algebraic structure of its domain.

Definition: Let (G , * ) and (G ,* ) be two groups. A mapping f : G — G is said to be a group
1
2
2
1
2
1
homomorphism (or just a homomorphism), if
f(x * y) = f(x) * f(y)  x, y  G .
1 2 1
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