Discrete Mathematics - Vol 3 | 13. Group Theory by Abraham | Learn Smarter
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Knowledge
13. Group Theory

The chapter provides a foundational understanding of group theory within abstract algebra, emphasizing the key axioms that define a group along with various examples. It covers fundamental properties such as closure, associativity, identity, and inverses for different sets and operations, and introduces the concept of abstract groups and the relevance of group properties in broader applications, such as computer science and cryptography.

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Sections

  • 13

    Group Theory

    Group Theory explores the definition and properties of mathematical groups, emphasizing the four axioms required for a set with a binary operation to be classified as a group.

    Learning Practice

  • 13.1

    Definition Of Groups

    This section introduces the fundamental concepts of group theory, defining a group and its axioms.

    Learning Practice

  • 13.2

    Group Axioms

    This section introduces the key axioms defining groups in abstract algebra.

    Learning Practice

  • 13.2.1

    Closure Property

    The closure property is a fundamental aspect of groups in algebra, asserting that the result of an operation on any two elements in a set must also belong to that set.

    Learning Practice

  • 13.2.2

    Associativity Property

    The Associativity Property in group theory states that the order of operations does not affect the outcome when combining elements of a group.

    Learning Practice

  • 13.2.3

    Identity Element

    This section defines the identity element in group theory and its critical role within mathematical structures known as groups.

    Learning Practice

  • 13.2.4

    Inverse Element

    The section elucidates the concept of inverse elements in group theory, providing comprehensive properties and definitions necessary for understanding groups.

    Learning Practice

  • 13.3

    Examples Of Groups

    This section introduces group theory, detailing group definitions and their properties through examples.

    Learning Practice

  • 13.3.1

    Group Of Integers Under Addition

    This section introduces groups in abstract algebra, focusing on the group of integers under addition, explaining key group properties.

    Learning Practice

  • 13.3.2

    Non-Negative Integers Under Addition

    This section discusses the concept of groups in mathematics, specifically focusing on non-negative integers under addition and the properties required to form a group.

    Learning Practice

  • 13.3.3

    Group Of Non-Zero Real Numbers Under Multiplication

    This section discusses the properties of groups, particularly focusing on the group of non-zero real numbers under multiplication.

    Learning Practice

  • 13.3.4

    Non-Zero Integers Under Multiplication

    This section discusses the properties of groups, particularly focusing on the non-zero integers under multiplication, and why this set does not form a group.

    Learning Practice

  • 13.4

    Addition Modulo K

    This section discusses the concept of groups in abstract algebra, specifically focusing on the properties of groups and the example of addition modulo k.

    Learning Practice

  • 13.5

    Multiplication Modulo K

    This section introduces the concept of multiplication modulo k, exploring its definition, properties, examples, and establishing it as a group operation.

    Learning Practice

  • 13.6

    Abstract Groups

    This section introduces the concept of abstract groups in group theory, defining the essential properties that constitute a group.

    Learning Practice

  • 13.6.1

    Abelian Groups

    Abelian groups are a special class of groups in which the group operation is commutative.

    Learning Practice

  • 13.6.2

    Group Order

    This section introduces the concept of groups in abstract algebra, detailing the group axioms and providing various examples.

    Learning Practice

References

ch62.pdf

Class Notes

Memorization

What we have learnt

  • Groups are defined by four ...
  • Different sets, such as int...
  • Abstract algebra enables th...

Final Test

Revision Tests