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Abstract Algebra: Group Theory

N.B. Singh

This audiobook is narrated by a digital voice. "Abstract Algebra: Group Theory" is an introductory textbook designed for absolute beginners seeking a foundational understanding of abstract algebra through the lens of group theory. This comprehensive...

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Preface

1/19/2025
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About This Book

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Introduction to Group Theory

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Definitions and Examples

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Basic Properties of Groups

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Finite and Infinite Groups

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Symmetry and Permutation Groups

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Matrix Groups

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Dihedral Groups

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Historical Background

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Subgroups and Cyclic Groups

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Subgroup Criteria

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Examples of Subgroups

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Cyclic Groups and Generators

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Properties of Cyclic Groups

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Order of an Element

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Subgroups of Cyclic Groups

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Applications of Cyclic Groups

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Group Homomorphisms and Isomorphisms

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Kernel and Image

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Properties of Homomorphisms

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The Isomorphism Theorems

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Automorphisms

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Endomorphisms

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Applications of Homomorphisms

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Cosets and Lagrange’s Theorem

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Cosets and Their Properties

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Lagrange’s Theorem

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Applications of Lagrange’s Theorem

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Index of a Subgroup

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Normalizers and Centralizers

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The Counting Principle

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Applications in Symmetry

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Normal Subgroups and Quotient Groups

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Normal Subgroups: Definition and Examples

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Properties of Normal Subgroups

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Quotient Groups

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The First Isomorphism Theorem

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The Second and Third Isomorphism Theorems

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The Correspondence Theorem

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Applications of Quotient Groups

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Group Actions and Sylow Theorems

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Group Actions: Definition and Examples

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Transitive Group Actions

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The Orbit-Stabilizer Theorem

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Applications of Group Actions

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Sylow Theorems: Introduction

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Proofs of Sylow Theorems

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Applications of Sylow Theorems

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Advanced Topics in Group Theory

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Solvable and Nilpotent Groups

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Free Groups

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Group Presentations

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Commutator Subgroups

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Group Extensions

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Simple Groups

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Recent Developments in Group Theory

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