Algebra For B.Sc. 3rd Sem.Kanha

Algebra is a meticulously crafted textbook for B.Sc. 2nd Year 3rd Semester students, authored by distinguished scholars Dr. K.B. Gupta, Dr. A.K. Jaiswal, Dr. C.S. Singh, Dr. V.S. Chaubey, and Dr. Kiruba Seeni Anantham. Published by Kanha Publishing House, Gorakhpur, this 200-page book provides a comprehensive and detailed study of algebraic concepts, essential for a solid understanding of higher mathematics.

Authors’ Background:

• Dr. K.B. Gupta (St. Andrew’s P.G. College, Gorakhpur)
• Dr. A.K. Jaiswal (St. Andrew’s P.G. College, Gorakhpur)
• Dr. C.S. Singh (Budha P.G. College, Kushinagar)
• Dr. V.S. Chaubey (B.R.D. P.G. College, Deoria)
• Dr. Kiruba Seeni Anantham (St. Joseph’s College For Women, Gorakhpur)

Contents Overview:

Unit I:

1. Properties of Integers, Divisor, Division Algorithm
   • Detailed explanation of integers, divisors, and the division algorithm.
2. Greatest Common Divisor, Euclidean Algorithm
   • In-depth study of the greatest common divisor and the Euclidean algorithm.
3. Fundamental Theorem of Arithmetic
   • Comprehensive analysis of the fundamental theorem of arithmetic.
4. Congruences and Residue Classes
   • Detailed study of congruences and residue classes.
5. Euler’s φ-Function and Its Properties
   • Exploration of Euler’s φ-function and its properties.
6. Euler’s, Fermat’s, and Wilson’s Theorem
   • In-depth analysis of Euler’s, Fermat’s, and Wilson’s theorems.

Unit II:

1. Algebraic Structure
   • Definition and examples of algebraic structures.
2. Definition of a Group with Examples and Simple Properties
   • Introduction to groups, with examples and their basic properties.
3. Subgroups, Generators of a Group, Cyclic Groups
   • Detailed study of subgroups, group generators, and cyclic groups.
4. Order of an Element of a Group, Centre of Group
   • Analysis of the order of group elements and the center of a group.

Unit III:

1. Permutation Groups
   • Comprehensive study of permutation groups.
2. Cyclic Permutation, Transposition, Even and Odd Permutations
   • Exploration of cyclic permutations, transpositions, and even and odd permutations.
3. The Alternating Group, Cayley’s Theorem
   • In-depth study of the alternating group and Cayley’s theorem.
4. Direct Products, Coset Decomposition
   • Detailed analysis of direct products and coset decomposition.
5. Lagrange’s Theorem and Its Consequences
   • Thorough exploration of Lagrange’s theorem and its implications.

Unit IV:

1. Homomorphism and Isomorphism
   • Detailed study of homomorphisms and isomorphisms in algebra.
2. Kernel of Homomorphism, Normal Subgroups, Simple Groups
   • Exploration of the kernel of homomorphisms, normal subgroups, and simple groups.
3. Quotient Groups, Fundamental Theorem of Homomorphism
   • Comprehensive analysis of quotient groups and the fundamental theorem of homomorphisms.
4. Theorems on Isomorphism
   • Detailed study of various theorems on isomorphism.

Purpose and Audience:

Algebra is designed to provide a thorough understanding of algebraic principles and structures, essential for academic success and future research in mathematics. This book is an indispensable resource for both students and educators aiming to master higher-level algebra.