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Page 1
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Mathematics
Lesson: Lagrange's Theorem and Homomorphism
Course Developer: Umesh Chand
Department / College: Assistant Professor, Department of
Mathematics, Kirorimal College
University of Delhi
Page 2
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Mathematics
Lesson: Lagrange's Theorem and Homomorphism
Course Developer: Umesh Chand
Department / College: Assistant Professor, Department of
Mathematics, Kirorimal College
University of Delhi
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Lagrange's theorem and its consequences, Homomorphism,
Properties of Homomorphism
? 1. Learning outcomes
? 2. Introduction
? 3. Subgroup
? 4. Coset
? 5. Lagrange's theorem.
? 6. Normal subgroup
? 7. Quotient group
? 8. Homomorphism.
8.1 Kernel of homomorphism.
8.2 Properties of homomorphism.
8.3 Properties of subgroups under homomorphism.
? Exercise
? Summary
? References / Bibliography / Further Reading
Page 3
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Mathematics
Lesson: Lagrange's Theorem and Homomorphism
Course Developer: Umesh Chand
Department / College: Assistant Professor, Department of
Mathematics, Kirorimal College
University of Delhi
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Lagrange's theorem and its consequences, Homomorphism,
Properties of Homomorphism
? 1. Learning outcomes
? 2. Introduction
? 3. Subgroup
? 4. Coset
? 5. Lagrange's theorem.
? 6. Normal subgroup
? 7. Quotient group
? 8. Homomorphism.
8.1 Kernel of homomorphism.
8.2 Properties of homomorphism.
8.3 Properties of subgroups under homomorphism.
? Exercise
? Summary
? References / Bibliography / Further Reading
Lagrange's Theorem and Homomorphism
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
? Lagrange's theorem.
? Consequences of Lagrange's theorem.
? Homomorphism and its properties
2. Introduction:
The Lagrange's theorem, Homomorphism are the most important
concept in Algebra. The consequences of Lagrenge's theorem is that the
order of any element of finite group, divides the order of group. This can
be used to prove any another theorem as Fermat's little theorem and
Euler's theorem.
The homomorphism is also very useful in modern algebra; the word
homomorphism come from the ancient Greek word; homo meaning
"same" and morphe meaning "shape".
3. Subgroup: Let G be a group. A non empty subset H of a group G
is said to be subgroup of G if H form the group under the binary operation
of G.
Example 1: Set of all even integers E with "+" is a subgroup of (Z, +).
Value addition : Note
1. A non empty subset H of a group G is a subgroup of G iff
(i) a, b ? H ? ab ? H
(ii) a ? H ? a
?1
?
H.
2. A non empty subset H of a group G is subgroup of G iff ab
?1
? H ?
Page 4
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Mathematics
Lesson: Lagrange's Theorem and Homomorphism
Course Developer: Umesh Chand
Department / College: Assistant Professor, Department of
Mathematics, Kirorimal College
University of Delhi
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Lagrange's theorem and its consequences, Homomorphism,
Properties of Homomorphism
? 1. Learning outcomes
? 2. Introduction
? 3. Subgroup
? 4. Coset
? 5. Lagrange's theorem.
? 6. Normal subgroup
? 7. Quotient group
? 8. Homomorphism.
8.1 Kernel of homomorphism.
8.2 Properties of homomorphism.
8.3 Properties of subgroups under homomorphism.
? Exercise
? Summary
? References / Bibliography / Further Reading
Lagrange's Theorem and Homomorphism
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
? Lagrange's theorem.
? Consequences of Lagrange's theorem.
? Homomorphism and its properties
2. Introduction:
The Lagrange's theorem, Homomorphism are the most important
concept in Algebra. The consequences of Lagrenge's theorem is that the
order of any element of finite group, divides the order of group. This can
be used to prove any another theorem as Fermat's little theorem and
Euler's theorem.
The homomorphism is also very useful in modern algebra; the word
homomorphism come from the ancient Greek word; homo meaning
"same" and morphe meaning "shape".
3. Subgroup: Let G be a group. A non empty subset H of a group G
is said to be subgroup of G if H form the group under the binary operation
of G.
Example 1: Set of all even integers E with "+" is a subgroup of (Z, +).
Value addition : Note
1. A non empty subset H of a group G is a subgroup of G iff
(i) a, b ? H ? ab ? H
(ii) a ? H ? a
?1
?
H.
2. A non empty subset H of a group G is subgroup of G iff ab
?1
? H ?
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 4
a,b ? H.
4. Coset: Let G be a group. H is a subgroup of G. For any a ? G, the set
{ah | h ? H} is denoted by aH, is called left coset of H in G by a.
Similarly the set {ha | h ? H} is denoted by Ha, is called right coset of H
in G by a.
| Ha | denote the number of elements in the set Ha.
Example 2: Let G = S
3
and H = {(1), (13)}, then the all left-cosets of H
in G are
(1) H = H
(12) H = {(12), (12)(13)} = {(12), (132)} = (132)H
(13)H = {(13), {1)} = H
(2, 3)H = {(23), (23), (13)} = {(23), (123)} = (123)H
Value addition : Remember
Let H be a subgroup of a group G, and let a and b belongs to G.
Then
(i) a ? aH
(ii) aH = H iff a ? H
(iii) aH = bH or aH ? bH = ?
(iv) aH = bH iff a
?1
b ? H
(v) | aH | = | bH |
(vi) aH = Ha iff H = aHa
?1
(vii) aH is a subgroup of G iff a ?H.
Page 5
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 1
Subject: Mathematics
Lesson: Lagrange's Theorem and Homomorphism
Course Developer: Umesh Chand
Department / College: Assistant Professor, Department of
Mathematics, Kirorimal College
University of Delhi
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 2
Table of Contents:
Chapter: Lagrange's theorem and its consequences, Homomorphism,
Properties of Homomorphism
? 1. Learning outcomes
? 2. Introduction
? 3. Subgroup
? 4. Coset
? 5. Lagrange's theorem.
? 6. Normal subgroup
? 7. Quotient group
? 8. Homomorphism.
8.1 Kernel of homomorphism.
8.2 Properties of homomorphism.
8.3 Properties of subgroups under homomorphism.
? Exercise
? Summary
? References / Bibliography / Further Reading
Lagrange's Theorem and Homomorphism
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
? Lagrange's theorem.
? Consequences of Lagrange's theorem.
? Homomorphism and its properties
2. Introduction:
The Lagrange's theorem, Homomorphism are the most important
concept in Algebra. The consequences of Lagrenge's theorem is that the
order of any element of finite group, divides the order of group. This can
be used to prove any another theorem as Fermat's little theorem and
Euler's theorem.
The homomorphism is also very useful in modern algebra; the word
homomorphism come from the ancient Greek word; homo meaning
"same" and morphe meaning "shape".
3. Subgroup: Let G be a group. A non empty subset H of a group G
is said to be subgroup of G if H form the group under the binary operation
of G.
Example 1: Set of all even integers E with "+" is a subgroup of (Z, +).
Value addition : Note
1. A non empty subset H of a group G is a subgroup of G iff
(i) a, b ? H ? ab ? H
(ii) a ? H ? a
?1
?
H.
2. A non empty subset H of a group G is subgroup of G iff ab
?1
? H ?
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 4
a,b ? H.
4. Coset: Let G be a group. H is a subgroup of G. For any a ? G, the set
{ah | h ? H} is denoted by aH, is called left coset of H in G by a.
Similarly the set {ha | h ? H} is denoted by Ha, is called right coset of H
in G by a.
| Ha | denote the number of elements in the set Ha.
Example 2: Let G = S
3
and H = {(1), (13)}, then the all left-cosets of H
in G are
(1) H = H
(12) H = {(12), (12)(13)} = {(12), (132)} = (132)H
(13)H = {(13), {1)} = H
(2, 3)H = {(23), (23), (13)} = {(23), (123)} = (123)H
Value addition : Remember
Let H be a subgroup of a group G, and let a and b belongs to G.
Then
(i) a ? aH
(ii) aH = H iff a ? H
(iii) aH = bH or aH ? bH = ?
(iv) aH = bH iff a
?1
b ? H
(v) | aH | = | bH |
(vi) aH = Ha iff H = aHa
?1
(vii) aH is a subgroup of G iff a ?H.
Lagrange's Theorem and Homomorphism
Institute of Lifelong Learning, University of Delhi pg. 5
Value addition: Note
1. There is always a one-one onto mapping between any two right
coset of H in G.
2. The number of distinct left (or right) cosets of H in G is given by
G
O(G)
i (H)
O(H)
?? No. of distinct left cosets
It is called index of subgroup H of a group G.
5. Lagrange's theorem:
Theorem 1: Let G be a finite group and H is a subgroup of G then O(H) |
O(G).
Proof: Let
1 2 r
a H, a H,...a H denote the distinct left coset of H in G.
Let aG ? be any element of G, then a ? a
i
H for some i
?
1 2 r
a a H a H.... a H ? ? ?
?
1 2 r
G a H a H.... a H ? ? ?
Also
1 2 r
a H a H.... a H G ? ? ?
?
1 2 r
G a H a H.... a H ? ? ?
Since
1 2 r
G a H, a H....a H ? are distinct left coset
1 2 r
O(G) O(a H) (a H).... O(a H) ? ? ?
O(H) (H).... O(H) ? ? ? [ O(aH) O(H)] ? ?
? O(G) rO(H) ?
Hence O(H) | O(G).
Read More
FAQs on Lecture 3 - Lagrange's Theorem and Homomorphism - Group Theory- Definition, Properties - Engineering Mathematics
| 1. What is Lagrange's Theorem? |  |
Ans. Lagrange's Theorem, also known as the Lagrange's Remainder Theorem, is a mathematical theorem that provides an estimate of the remainder when a function is approximated using its Taylor polynomial. It states that if a function f(x) is infinitely differentiable on an interval I that contains the point a, and if all its derivatives up to the nth derivative are continuous on I, then the remainder Rn(x) when the nth degree Taylor polynomial is used to approximate f(x) is given by the formula Rn(x) = f(n+1)(c)(x-a)^(n+1)/(n+1), where c is some number between x and a.
| 2. What is homomorphism in mathematics? | |
Ans. In mathematics, homomorphism is a structure-preserving map or function between two algebraic structures of the same type. It is a concept used in various branches of mathematics, such as group theory, ring theory, and module theory. A homomorphism between two structures preserves the operations and relations defined on the structures, meaning that if we apply the operations to elements in one structure and then map them using the homomorphism, it will result in the same outcome as applying the operations to the corresponding elements in the other structure.
| 3. How can Lagrange's Theorem be applied in engineering mathematics? | |
Ans. Lagrange's Theorem has applications in various fields of engineering mathematics. It is commonly used in numerical analysis and approximation theory to estimate the error or remainder when a function is approximated using its Taylor polynomial. This is particularly useful in engineering calculations where accurate approximations of functions are needed. By using Lagrange's Theorem, engineers can determine the accuracy of their approximations and make adjustments accordingly, ensuring the reliability of their calculations.
| 4. Can Lagrange's Theorem be used to approximate any function? | |
Ans. Lagrange's Theorem can be applied to approximate any function that satisfies the required conditions. However, it is important to note that the accuracy of the approximation depends on the behavior of the function and the interval on which it is defined. If a function is not infinitely differentiable or if its derivatives are not continuous on the interval of interest, then the approximation using Lagrange's Theorem may not be accurate. Engineers and mathematicians need to carefully analyze the properties of the function before applying Lagrange's Theorem for approximation purposes.
| 5. How does homomorphism engineering contribute to practical applications? | |
Ans. Homomorphism engineering plays a significant role in various practical applications. In computer science and cryptography, homomorphic encryption schemes are used to perform computations on encrypted data without decrypting it, ensuring privacy and security. Homomorphic image processing techniques allow for the manipulation of encrypted images without revealing their contents. Homomorphism is also utilized in signal processing, where it enables the analysis and manipulation of signals while preserving their essential characteristics. In summary, homomorphism engineering provides powerful tools for secure computations and signal processing in practical applications.
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