Page 89 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra Sachin Kaushal, Lovely Professional University

Notes Unit 7: Homomorphism Theorem

CONTENTS
Objectives

Introduction
7.1 Fundamental Theorem of Homomorphism
7.2 Automorphisms
7.3 Summary
7.4 Keywords

7.5 Review Questions
7.6 Further Readings

Objectives

After studying this unit, you will be able to:
Discuss fundamental theorem of homomorphism

Explain the concept of automorphism

Introduction

After understanding the concept of isomorphisms. Let us prove some result about the relationship
between homomorphisms and quotient groups. The first result is the Fundamental Theorem of
Homomorphism for groups. It is called ‘fundamental’ because a lot of group theory depends
upon this result. This result is also called the first isomorphism theorem.

7.1 Fundamental Theorem of Homomorphism

Theorem 1 (Fundamental Theorem of Homomorphism): Let G and G be two groups and f : G 1
2
1
 G be a group homomorphism. Then
2
G /Ker f  Im f.
1
In particular, if f is onto, then G /Ker f  G . 2
1
Proof: Let Ker f = H. Note that H  G . Let us define the function
1
 : G /H  Im f :  (Hx) = f(x).
1
At first glance it seems that the definition of  depends on the coset representative. But we
will show that if x, y  G such that Hx = Hy, then (Hx) = (Hy). This will prove that  is a
1
well-defined function.
Now, Hx = Hy  xy  H = Ker f  f(xy ) = e , the identity of G .
-1
-1
2
2
 f(x)[f(y)] = e  f(x) = f(y).
-1
2
 (Hx) = (HY).

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