Basics of Modern Abstract Algebra

Description

Abstract algebra is one of the fundamental branches of mathematics. It provides a systematic framework for studying algebraic structures and the relationships among them. The concepts introduced in this subject are essential not only for advanced mathematics but also for applications in computer science, coding theory, cryptography, physics and other scientific disciplines.

This book, Basics of Modern Abstract Algebra, has been prepared primarily for undergraduate students studying mathematics under the UGC NEP 2020 curriculum. It is intended to provide a clear, logically organized and accessible introduction to the basic concepts of abstract algebra. The presentation begins with elementary ideas and gradually develops the subject through definitions, examples, theorems, proofs and exercises.

The book is organized into four chapters. Chapter 1 introduces sets, subsets, mappings, families of sets, equivalence relations and the properties of integers required for the subsequent study of algebra. Chapter 2 develops group theory, including subgroups, cosets, homomorphisms, permutation groups, symmetric groups, direct products, semidirect products and finite Abelian groups. Chapter 3 discusses vector spaces, linear transformations, linear independence, bases and dual spaces. Chapter 4 presents rings and modules, including quotient rings, fields of quotients, principal ideal domains, polynomial rings and modules over principal ideal domains.

The book may be used as a classroom textbook, a reference source or a guide for independent study. It is hoped that the systematic arrangement of the material will help students develop mathematical maturity and prepare them for further study in algebra and related areas.

I express my sincere gratitude to my teachers, colleagues and students, whose suggestions and encouragement contributed to the preparation of this book. I also welcome constructive comments from readers that may help improve future editions.