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Abstract Algebra is a mathematical field that explores algebraic structures such as groups, rings, and fields, focusing on their properties and relationships. Within this discipline, Group Theory specifically studies group sets with an operation that satisfies certain axioms. This area of study provides essential insights into concepts like symmetry and transformations, finding applications in various domains, including physics, chemistry, and computer science. Through the exploration of group theory, students discover the foundational structures that govern a wide range of mathematical systems.
What Will I Learn?
- Fundamental Concepts: Understand the basic principles of algebraic structures, including groups, rings, and fields.
- Properties and Axioms: Explore the defining properties and axioms that characterize these structures.
- Operations: Gain insights into operations, homomorphisms, and isomorphisms within algebraic systems.
- Key Topics: Delve into important subjects such as the order of groups, group actions, and symmetry.
- Applications: Discover the applications of group theory in various fields, including physics and computer science.
- Critical Thinking Skills: Develop analytical and problem-solving skills to better understand complex mathematical systems.
Targeted Audience
- Students: High school and college learners pursuing mathematics or related disciplines who aim to enhance their grasp of abstract algebra and group theory.
- Educators: Teachers and professors seek resources and materials to enrich their algebra and group theory curriculum.
- Lifelong Learners: Individuals with a strong interest in mathematics who wish to delve into advanced concepts and broaden their knowledge beyond conventional studies.
- Researchers: Mathematicians and professionals in related fields looking to apply group theory and abstract algebra in their research and work.
- Enthusiasts: Anyone fascinated by the elegance and applications of mathematics, eager to engage with complex mathematical ideas.
Course materials
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1
Lecture 1:This lesson describes Historical note of group Theory
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2
Lecture 2
In This Lesson we Describes about Groups and Properties of Group
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3
Lecture 3:
Torsion-Free and Mixed Groups:
This lesson explores torsion-free groups, which have no elements of finite order, alongside mixed groups that feature both torsion and torsion-free elements, showcasing their distinct properties in group theory.
Semi-Groups and Monoids:
In this lesson, we investigate semi-groups characterized by associative binary operations and monoids that incorporate an identity element, highlighting their fundamental significance in algebra and their applications across various domains.
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4
Lecture 4
In This Lesson We describes about Abelian group and its Examples
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5
Lecture 5
we will Describes about Historical Note about an abelian group or commutative
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6
Lecture 6
In This Lesson we will discuss about
Order of a Group
Order of an element
Periodic Group
Involution:
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7
Lecture 7:
In This Lession We Will Discuss About :
Subgroup
Proper and Improper subgroups
Examples of sub-groups:
Theorem
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8
Lecture 8
This Lesson Describes:
Cyclic group :
Results on cyclic groups
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9
Lecture 9
This Lesson Describes:
Centre of a group
Results
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10
Lecture 10
we will Briefly Describes about:
Klein’s four groups
Properties
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11
Lecture 11
In This Lesson we will discribe about :
Normalizer of a Subgroup
Results:
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12
Lecture 12
In This Lesson We Will Discuss About:
First Isomorphism Theorem
Second Isomorphism Theorem
Third Isomorphism Theorem
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13
Lecture 13
In This Lesson WE will Learn About :
Complete Group
Derived Group OR Commutator Subgroup:
Simple group :
Direct Product:
Properties:
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14
Lecture 14
In This Lesson We Will Discuss About :
Quaternion Group
Properties
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15
Lecture 15
In This Lesson We Will Discuss About :
Homomorphism
Endomorphism
Monomorphism
Epimorphism
Isomorphism
Theorem
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16
Lecture 16
In This Lesson WE Will Learn About:
Conjugation as an Automorphism:
Embedding:
Simple group :
Complete Group:
Derived Group OR Commutator Subgroup:
Commutator subgroup of D3:
Cauchy theorem for finite abelian groups :
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17
Lecture 17
In This Lesson We Will Learn About :
Normal subgroups:
Examples
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