Programmes → I.M.Sc. in Mathematical Sciences; M.Sc. in Mathematics/Applied Mathematics/Statistics
Algebra-I
Course Code: MA 303; IM 303; ST 303; MM 403 | Credits: 4 | Prerequisite: The students must have done at least an algebra course in their Bachelor's degree.
Syllabus Details
- Unit I: Sets, relations, functions, equivalence relations, equivalence classes, partial order relations, posets, chains, anti-chains, Zorn’s lemma, lub and glb in a poset, Lattice, distributive lattice, universal upper/lower bonds, complemented distributive Lattices, Boolean lattices/algebra, Finite Boolean lattices.
- Unit II: Groups and Rings: Introduction, Semi-groups, Groups, Subgroup, Generators and Evaluation of Powers, Cosets and Lagrange’s Theorem, Permutation Groups, Normal Subgroups, Quotient groups.
- Unit III: Group homomorphisms, Automorphisms, Isomorphisms, Fundamental theorems of group homomorphisms, Cayley’s Theorem, Group actions, Burnside’s Theorem, Sylow’s Theorems 1st, 2nd, 3rd and their applications.
- Unit IV: Structure theorem for finite Abelian groups, Composition series, Jordan-Holder theorem, Nilpotent and solvable groups.
Course Learning Objective
- Understand a Group.
- Understand and solve some basic problems.
- Apply these concepts for further study in Algebra.
Recommended Books
- Michael Artin, Algebra, 2nd ed. Pearson, Upper Saddle River, NJ, 2011
- I. N. Herstein, Topics in Algebra, 3rd ed. Wiley, New York, 1996
- Dummit and Foote, Abstract Algebra, 3rd ed. Wiley, New York, 2003
- P. R. Halmos, Naive Set Theory, Springer, New York, 1991