Algebra I | Department of Mathematics

Algebra I

Core course for B.Sc. (Research) Mathematics.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII Mathematics or MAT100 (Foundations)

Overview: Algebraic structures like groups, rings, integral domains, fields, modules and vector spaces are present in almost all mathematical applications as well as in development of more complicated structures in mathematics. The basic building block of these structures is the group. This, a first course in Abstract Algebra, concentrates mainly on groups and their basic properties. If there is time, we shall also take a brief look at Rings.

It is desirable that the student already has a basic understanding of sets, relations, functions, binary operations, equivalence relations, and sets of numbers.

Detailed Syllabus:

  1. Groups: Definition, Examples and Elementary Properties.
    Subgroups: Subgroup Tests, Subgroups Generated by Sets, Cyclic Groups, Classification of Subgroups of Cyclic Groups, Cosets and Lagrange's Theorem.
  2. Normal Subgroups and Quotient Groups, Homomorphisms, Isomorphisms and Automorphisms of a Group.
    Conjugates, centre, centralizer, normalizer.
    Cayley’s Theorem.
    Direct Products, Finite Abelian Groups.
  3. Permutation Groups: Definition, Examples and Properties, Symmetric Group of n Letters (Sn), Alternating Group on n Letters (An).
  4. (If time permits) Rings, Homomorphisms, Ideals and Quotient Rings, Integral Domains.

References:

  1. Contemporary Abstract Algebra by Joseph A. Gallian, 4th  edition. Narosa, 1999.
  2. Algebra by Michael Artin, 2nd Edition. Prentice Hall India, 2011.
  3. Topics in Algebra by I.N. Herstein, 2nd Edition. Wiley India, 2006.
  4. A First Course in Abstract Algebra by John B. Fraleigh, 7th Edition. Pearson, 2003.
  5. Undergraduate Algebra by Serge Lang, 2nd Edition. Springer India, 2009.
  6. Abstract Algebra by David S. Dummit and Richard M. Foote, 3rd Edition. John Wiley and Sons, 2011.

Past Instructors: Neha Gupta, Sanjeev Agrawal

Course Code: 
MAT240
Course Credits: 
4.00
Department: 
Course Level: