Lecture 4 - Isomorphism and Theorems on Isomorphism | Group Theory- Definition, Properties - Engineering Mathematics PDF Download

 Page 1


Isomorphism and Theorems on Isomorphism 
Institute of Lifelong Learning, University of Delhi                                                  pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Isomorphism and Theorems on Isomorphism 
Lesson Developer: Umesh Chand 
Department / College: Assistant Professor, Department 
of Mathematics, Kirorimal College 
University of Delhi 
 
 
 
 
 
 
 
  
Page 2


Isomorphism and Theorems on Isomorphism 
Institute of Lifelong Learning, University of Delhi                                                  pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Isomorphism and Theorems on Isomorphism 
Lesson Developer: Umesh Chand 
Department / College: Assistant Professor, Department 
of Mathematics, Kirorimal College 
University of Delhi 
 
 
 
 
 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
Chapter: Isomorphism and Theorems on Isomorphism 
? 1. Learning outcomes 
? 2. Introduction 
? 3.  Isomorphism. 
3.1  Properties of isomorphism. 
? 4. External direct product of groups 
? 5. Automorphism 
? Exercise 
? Summary 
? References / Bibliography / Further Reading. 
 
  
Page 3


Isomorphism and Theorems on Isomorphism 
Institute of Lifelong Learning, University of Delhi                                                  pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Isomorphism and Theorems on Isomorphism 
Lesson Developer: Umesh Chand 
Department / College: Assistant Professor, Department 
of Mathematics, Kirorimal College 
University of Delhi 
 
 
 
 
 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
Chapter: Isomorphism and Theorems on Isomorphism 
? 1. Learning outcomes 
? 2. Introduction 
? 3.  Isomorphism. 
3.1  Properties of isomorphism. 
? 4. External direct product of groups 
? 5. Automorphism 
? Exercise 
? Summary 
? References / Bibliography / Further Reading. 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning outcomes 
After studying the whole content of this chapter, students will be able to understand 
? Isomorphism  
? Properties of Isomorphism. 
? Fundamental theorems of homomorphism. 
? Automorphism.  
 
2. Introduction 
 Isomorphism is most important concept in Algebra. It is an extension of 
homomorphism. With the help of isomorphism, we realizes that two or more groups 
defined in different term are really the same or not. The term isomorphism is derived 
from the Greek words isos, "same" or "equal" and morphs "form". 
 
 
  
Page 4


Isomorphism and Theorems on Isomorphism 
Institute of Lifelong Learning, University of Delhi                                                  pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Isomorphism and Theorems on Isomorphism 
Lesson Developer: Umesh Chand 
Department / College: Assistant Professor, Department 
of Mathematics, Kirorimal College 
University of Delhi 
 
 
 
 
 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
Chapter: Isomorphism and Theorems on Isomorphism 
? 1. Learning outcomes 
? 2. Introduction 
? 3.  Isomorphism. 
3.1  Properties of isomorphism. 
? 4. External direct product of groups 
? 5. Automorphism 
? Exercise 
? Summary 
? References / Bibliography / Further Reading. 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning outcomes 
After studying the whole content of this chapter, students will be able to understand 
? Isomorphism  
? Properties of Isomorphism. 
? Fundamental theorems of homomorphism. 
? Automorphism.  
 
2. Introduction 
 Isomorphism is most important concept in Algebra. It is an extension of 
homomorphism. With the help of isomorphism, we realizes that two or more groups 
defined in different term are really the same or not. The term isomorphism is derived 
from the Greek words isos, "same" or "equal" and morphs "form". 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 4 
 
3.  Isomorphism 
 An isomorphism from a group G to a group G ? is one to one mapping from G to G ? 
that preserved the group operation. 
Value Addition: Note 
Two groups G and G ? are called isomorphic, written as G
1
 ~ G ?
2
. If there is an 
isomorphism from Gonto G' i.e. two groups g and g ? are isomorphic if there exist 
a mapping ? : G ? G ? such that 
(i) ? is homomorphism. 
(ii) f is one-one 
(iii) f is onto.  
 
Example 1: Let G be the group of real numbers under addition and G ? be  the group of 
positive real numbers under multiplication. Then G and G ? are isomorphic under the 
mapping ?(x) = 2
x
. 
Solution: Since ? : G ? G ? is well defined. 
To Show: ? if one one. 
let ?(x) = ?(y) 
? 2
x
 = 2
y
 
? log
2
2
x
 = log
2
2
y
 
? x = y 
To Show: ? is onto. 
We must find for any positive real number y some real numbe x s.t.  
 ?(x) = y i.e. 2
x
 = y 
? x = log
2
y ? G 
? ? is onto. 
To show: ? is homomoirphism. 
For all x,y ? G 
 ?(x + y) = 2
x+y
 
Page 5


Isomorphism and Theorems on Isomorphism 
Institute of Lifelong Learning, University of Delhi                                                  pg. 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Isomorphism and Theorems on Isomorphism 
Lesson Developer: Umesh Chand 
Department / College: Assistant Professor, Department 
of Mathematics, Kirorimal College 
University of Delhi 
 
 
 
 
 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 2 
 
Table of Contents: 
Chapter: Isomorphism and Theorems on Isomorphism 
? 1. Learning outcomes 
? 2. Introduction 
? 3.  Isomorphism. 
3.1  Properties of isomorphism. 
? 4. External direct product of groups 
? 5. Automorphism 
? Exercise 
? Summary 
? References / Bibliography / Further Reading. 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 3 
 
1. Learning outcomes 
After studying the whole content of this chapter, students will be able to understand 
? Isomorphism  
? Properties of Isomorphism. 
? Fundamental theorems of homomorphism. 
? Automorphism.  
 
2. Introduction 
 Isomorphism is most important concept in Algebra. It is an extension of 
homomorphism. With the help of isomorphism, we realizes that two or more groups 
defined in different term are really the same or not. The term isomorphism is derived 
from the Greek words isos, "same" or "equal" and morphs "form". 
 
 
  
 
Institute of Lifelong Learning, University of Delhi                                                pg. 4 
 
3.  Isomorphism 
 An isomorphism from a group G to a group G ? is one to one mapping from G to G ? 
that preserved the group operation. 
Value Addition: Note 
Two groups G and G ? are called isomorphic, written as G
1
 ~ G ?
2
. If there is an 
isomorphism from Gonto G' i.e. two groups g and g ? are isomorphic if there exist 
a mapping ? : G ? G ? such that 
(i) ? is homomorphism. 
(ii) f is one-one 
(iii) f is onto.  
 
Example 1: Let G be the group of real numbers under addition and G ? be  the group of 
positive real numbers under multiplication. Then G and G ? are isomorphic under the 
mapping ?(x) = 2
x
. 
Solution: Since ? : G ? G ? is well defined. 
To Show: ? if one one. 
let ?(x) = ?(y) 
? 2
x
 = 2
y
 
? log
2
2
x
 = log
2
2
y
 
? x = y 
To Show: ? is onto. 
We must find for any positive real number y some real numbe x s.t.  
 ?(x) = y i.e. 2
x
 = y 
? x = log
2
y ? G 
? ? is onto. 
To show: ? is homomoirphism. 
For all x,y ? G 
 ?(x + y) = 2
x+y
 
 
Institute of Lifelong Learning, University of Delhi                                                pg. 5 
 
    = 2
x
2
y
 
  = ?(x). ?(y). 
So ? preserving the operation. 
Example 2. The mapping ? from R to R given by ?(x) = x
3
 is not isomorphism. Where 
group R is set of real numbers under addition. 
Solution: Since ?(x + y) = (x + y)
3
 ? x
3
 + y
3
 = ?(x) + ?(y) 
? ? is not homomorphism. 
Hence ? is not isomorphism.                                             
I. Q. 1 
Theorem 1: The relation ' ~ ' (relation of Isomorphism) is an equivalence relation. 
Proof: Let G* be the collection of all groups. 
To show :  ~ is an equivalence relation of G. 
Reflexive : To show G ~ G  ? G ? G
*
. 
Define a mapping g : G ? G as g(x) = x ? x ? G, this is identity mapping which is one-
one onto and homomorphism. 
Hence G ~ G. 
Symmetric: Let G, G ? ? G
*
 such that G ~ G ? therefore ? an isomorphism : G G ? ??  
? ? is one-one and onto ? ? is invertible i.e. ?
?1
 exist and is also one-one onto ? ?
?1
 : G ? 
? G is one-one and onto. 
To show: ?
?1
 is homomorphism 
i.e. 
1 1 1
(xy) (x) (y)
? ? ?
? ? ? ? 
let  
11
(x) a and (y) b
??
? ? ? ? 
? ?(a) = x  and ?(b) = y 
? xy = ?(a) ?(b)= ?(ab)    ( ? ? is homomorphism) 
? ?
?1
(xy) = ab = ?
?1
(x) ?
?1
(y)