Proof That R= (0,1,2,3,4,5,6) (R,+) Is An Abelian Group.​

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Introduction

In mathematics, a group is a set of elements combined under a binary operation that satisfies certain properties. One of the fundamental properties of a group is commutativity, also known as Abelian property. In this article, we will prove that the set R = (0,1,2,3,4,5,6) with the binary operation of addition (+) is an Abelian group.

What is an Abelian Group?

An Abelian group is a group that satisfies the commutative property, which means that the order of the elements in the group does not affect the result of the operation. In other words, for any two elements a and b in the group, a + b = b + a. This property is named after the Norwegian mathematician Niels Henrik Abel.

Properties of a Group

To prove that R = (0,1,2,3,4,5,6) is an Abelian group, we need to show that it satisfies the following properties:

  1. Closure: The result of the operation between any two elements in the group must be an element in the group.
  2. Associativity: The order in which we perform the operation between three elements does not affect the result.
  3. Identity Element: There must be an element in the group that does not change the result when combined with any other element.
  4. Inverse Element: For each element in the group, there must be another element that, when combined, results in the identity element.
  5. Commutativity: The order of the elements in the group does not affect the result of the operation.

Closure Property

To show that R = (0,1,2,3,4,5,6) satisfies the closure property, we need to demonstrate that the result of adding any two elements in the group is also an element in the group. Let's consider the following examples:

  • 0 + 1 = 1
  • 1 + 2 = 3
  • 2 + 3 = 5
  • 3 + 4 = 7 (which is not in the group, so we need to reconsider our group)
  • 4 + 5 = 9 (which is not in the group, so we need to reconsider our group)
  • 5 + 6 = 11 (which is not in the group, so we need to reconsider our group)

However, if we consider the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7, then we can see that:

  • 0 + 1 = 1
  • 1 + 2 = 3
  • 2 + 3 = 5
  • 3 + 4 = 7 ≡ 0 (mod 7)
  • 4 + 5 = 9 ≡ 2 (mod 7)
  • 5 + 6 = 11 ≡ 4 (mod 7)

Therefore, the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 satisfies the closure property.

Associativity Property

To show that R = (0,1,2,3,4,5,6) satisfies the associativity property, we need to demonstrate that the order in which we perform the operation between three elements does not affect the result. Let's consider the following examples:

  • (0 + 1) + 2 = 1 + 2 = 3
  • 0 + (1 + 2) = 0 + 3 = 3
  • (1 + 2) + 3 = 3 + 3 = 6
  • 1 + (2 + 3) = 1 + 5 = 6

Therefore, the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 satisfies the associativity property.

Identity Element

To show that R = (0,1,2,3,4,5,6) has an identity element, we need to find an element that does not change the result when combined with any other element. Let's consider the following examples:

  • 0 + 1 = 1
  • 1 + 0 = 1
  • 0 + 2 = 2
  • 2 + 0 = 2
  • ...

Therefore, the element 0 is the identity element of the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7.

Inverse Element

To show that R = (0,1,2,3,4,5,6) has an inverse element for each element, we need to find an element that, when combined with the given element, results in the identity element. Let's consider the following examples:

  • 1 + 6 = 0 (mod 7)
  • 2 + 5 = 0 (mod 7)
  • 3 + 4 = 0 (mod 7)
  • 4 + 3 = 0 (mod 7)
  • 5 + 2 = 0 (mod 7)
  • 6 + 1 = 0 (mod 7)

Therefore, each element in the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 has an inverse element.

Commutativity Property

To show that R = (0,1,2,3,4,5,6) satisfies the commutativity property, we need to demonstrate that the order of the elements in the group does not affect the result of the operation. Let's consider the following examples:

  • 0 + 1 = 1
  • 1 + 0 = 1
  • 0 + 2 = 2
  • 2 + 0 = 2
  • ...

Therefore, the group R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 satisfies the commutativity property.

Conclusion

In this article, we have shown that the set R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 satisfies the properties of a group, including closure, associativity, identity element, inverse element, and commutativity. Therefore, we can conclude that R = (0,1,2,3,4,5,6) is an Abelian group.

References

Introduction

In our previous article, we proved that the set R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7 is an Abelian group. In this article, we will answer some frequently asked questions about Abelian groups and provide additional insights into this mathematical concept.

Q: What is an Abelian group?

A: An Abelian group is a group that satisfies the commutative property, which means that the order of the elements in the group does not affect the result of the operation. In other words, for any two elements a and b in the group, a + b = b + a.

Q: What are the properties of an Abelian group?

A: An Abelian group must satisfy the following properties:

  1. Closure: The result of the operation between any two elements in the group must be an element in the group.
  2. Associativity: The order in which we perform the operation between three elements does not affect the result.
  3. Identity Element: There must be an element in the group that does not change the result when combined with any other element.
  4. Inverse Element: For each element in the group, there must be another element that, when combined, results in the identity element.
  5. Commutativity: The order of the elements in the group does not affect the result of the operation.

Q: What is an example of an Abelian group?

A: One example of an Abelian group is the set R = (0,1,2,3,4,5,6) with the binary operation of addition modulo 7. We proved this in our previous article.

Q: What is the difference between an Abelian group and a non-Abelian group?

A: The main difference between an Abelian group and a non-Abelian group is the commutative property. In an Abelian group, the order of the elements does not affect the result of the operation, whereas in a non-Abelian group, the order of the elements does affect the result of the operation.

Q: Can you give an example of a non-Abelian group?

A: One example of a non-Abelian group is the set S = (1,2,3) with the binary operation of matrix multiplication. We can see that the order of the elements affects the result of the operation:

  • (1,2,3) × (1,2,3) = (1,4,9)
  • (1,2,3) × (3,2,1) = (3,6,3)
  • (3,2,1) × (1,2,3) = (3,6,3)

Q: What are the applications of Abelian groups?

A: Abelian groups have many applications in mathematics and computer science, including:

  • Cryptography: Abelian groups are used in cryptographic protocols such as RSA and Diffie-Hellman key exchange.
  • Error-correcting codes: Abelian groups are used in error-correcting codes such as Reed-Solomon codes.
  • Graph theory: Abelian groups are used in graph theory to study the properties of graphs.
  • Number theory: Abelian groups are used in number theory to study the properties of integers.

Q: Can you recommend any resources for learning more about Abelian groups?

A: Yes, here are some resources for learning more about Abelian groups:

  • Books: "Group Theory" by David S. Dummit and Richard M. Foote, "Abstract Algebra" by David S. Dummit and Richard M. Foote.
  • Online courses: "Group Theory" on Coursera, "Abstract Algebra" on edX.
  • Websites: Wikipedia, MathWorld, Wolfram Alpha.

Conclusion

In this article, we answered some frequently asked questions about Abelian groups and provided additional insights into this mathematical concept. We hope that this article has been helpful in understanding Abelian groups and their applications.