Replete with thought-provoking examples, worked-out exercises and a new section on Automorphism of Groups, this fourth 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 exercises that include MCQs added at the end of each section.
Well-supported by precisely laid-out online resources, the book also provides interesting insights to the history of mathematics to enable readers understand the concepts in the right perspective.
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 in the Department of Mathematics, 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 Associate Professor and Head of the Department of Pure Mathematics, University of Calcutta.
About the Authors Preface to the Fourth 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
13. Ideals and Homomorphisms of Rings 13.1 Ideals and Quotient Rings 13.2 Homomorphisms 13.3 Maximal ideals and Prime ideals 13.4 Regular Rings 13.5 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 Answers Index