Syllabus

Group theory (25 hours)
Binary operations, Groups, Subgroups, Cyclic groups, Normal Subgroups and Quotient groups, Lagrange’s theorem, Centralizers, Normalizers, Homomorphisms, Isomorphism theorems, Symmetric groups, Simplicity of the Alternating group, Matrix groups, Direct products of groups, Fundamental theorem of finitely generated abelian groups, Sylow Groups.

Ring Theory (15 hours)
Rings, Integral Domains, Ideals, Ring of fractions, Ring homomorphisms, Polynomial rings, Fields.

Understand the fundamental concepts of group and ring theory.

Apply key theorems of abstract algebra, such as Lagrange's Theorem, the Isomorphism Theorems, and Cayley's Theorem, to solve mathematical problems.

Investigate group homomorphisms and quotient groups and understand their role in classifying algebraic structures.

Understand the notion of rings, integral domains and field via examples.

Explain the fundamental concepts of groups, subgroups, cyclic groups, permutations, rings, integral domains, and fields.

Apply important results such as Lagrange's Theorem, homomorphism theorems, and properties of ideals and quotient rings to solve abstract algebra problems.

Solve theoretical and computational problems involving finite groups, cyclic groups, polynomial rings, and finite fields using appropriate algebraic techniques.

Construct and analyze group homomorphisms, ring homomorphisms, normal subgroups, quotient groups, ideals, and factor rings.

I. N. Herstein, Abstract Algebra, 3rd Edition, John Wiley and Sons, 1996.

J. A. Gallian, Contemporary Abstract Algebra, 4th Edition, Narosa, 2019.

D. S. Dummit & R. M. Foote, Abstract Algebra, John Wiley and Sons, 2003.

J. J. Rotman, An Introduction to the Theory of Groups, Springer, 1995.