## Topics in Abstract Algebra, Third edition

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Replete with thought-provoking examples and worked-out exercises, this third edition of Topics in Abstract Algebra is designed in accordance with the new Choice Based Credit System (CBCS) syllabus of Abstract Algebra and Advanced Abstract Algebra prescribed by UGC for all Indian universities at the UG Honours/Advanced level. Students appearing for competitive entrance examinations such as NET, JAM, ISI, IISc., TIFR, NBHM, GATE, SET or MCA will benefit immensely from the carefully selected Solved Problems, about a hundred of which (of MCQ type) have been newly added to the present edition. Interesting insights to the history of mathematics and mathematicians are also provided in the course of discussion to enable the readers understand the concepts in the right perspective.

#### Contributors (Author(s), Editor(s), Translator(s), Illustrator(s) etc.)

Dr M K Sen, a noted algebraist, retired from the Department of Pure Mathematics, University of Calcutta, after a rich career spanning more than thirty years of teaching Abstract Algebra.

Dr Shamik Ghosh, a student of Dr Sen, is Professor and Head of the Mathematics Department, Jadavpur University.

Dr Parthasarathi Mukhopadhyay, also a student of Dr Sen, is Associate Professor at the Mathematics Department of Ramakrishna Mission Residential College, affiliated to the University of Calcutta.

Dr Sunil Kumar Maity, yet another student of Dr Sen, is Assistant Professor in the Department of Pure Mathematics, University of Calcutta.

Preface to the Third Edition
Preface to the First Edition

1. Preliminaries
1.1 Sets
1.2 Relations
1.3 Congruous Relation
1.4 Functions
1.5 Binary Operations

2. Integers
2.1 Divisibility
2.2 Mathematical Induction
2.3 Prime Numbers
2.4 Linear Diophantine Equations
2.5 Arithmetic Functions
2.6 More on Congruence

3. Groups
3.1 Elementary Properties

4. Permutation Groups

4.1 Permutation Group

5. Subgroups

5.1 Subgroups
5.2 Cyclic Groups
5.3 Cosets and Lagrange’s Theorem
5.4 Normal Subgroups and Quotient Groups

6. Homomorphisms of Groups

6.1 Homomorphisms
6.2 Isomorphism Theorems

7. Direct Product of Groups
7.1 Direct Product

8. Symmetry
8.1 Isometry
8.2 Four Isometries of the Plane
8.3 Group of Symmetries

9. Finite Abelian Groups
9.1 Finite Abelian Groups

10. Sylow Theorems
10.1 Group Actions
10.2 Class Equation and Cauchy’s Theorem
10.3 Sylow Theorems

11. Simple Groups
11.1 Simple Groups

12. Rings
12.1 Elementary Properties
12.2 Integral Domains, Division Rings and Fields
12.3 Subrings and Subfields
12.4 Regular Rings

13. Ideals and Homomorphisms of Rings
13.1 Ideals and Quotient Rings
13.2 Homomorphisms
13.3 Maximal ideals and Prime ideals
13.4 Embedding of Rings

14. Factorization in Integer Domains
14.1 Factorization Domain
14.2 Euclidean Domain

15. Polynomial Rings
15.1 Ring of Polynomials
15.2 Irreducibility of Polynomials

Appendices
Bibliography