21MAT216 MODERN ALGEBRA MATHEMATICS PAPER – IV-

Unit I-

Introduction to Groups, Symmetries of a Square, The Dihedral Groups, Definition and Examples of Groups, Elementary Properties of Groups, Finite Groups, Subgroups: Terminology and Notation, Subgroup Tests, Examples of Subgroups.

Unit II-

Cyclic Groups, Properties of Cyclic Groups, Classification of Subgroups of Cyclic Groups, Permutation Groups, Properties of Permutations, Isomorphisms: Definition and Examples, Cayley’s Theorem, Properties of Isomorphisms, Automorphisms.

Unit III-

Cosets and Lagrange’s Theorem, Application of Cosets to Permutation Groups, Normal Subgroups, Factor Groups, Applications of Factor Groups, Group Homomorphisms: Definition and Examples, Properties of Homomorphisms, The First Isomorphism Theorem.

Unit IV-

Rings, Motivation and Definition, Examples of Rings, Properties of Rings, Subrings, Integral Domains, Characteristic of a Ring.

Unit V-

Quotient Rings and Ideals, Homomorphism of rings and rings of polynomials.

Ideals, Factor Rings, Prime Ideals and Maximal Ideals, Ring Homomorphisms: Definition and Examples, Properties of Ring Homomorphisms, The Field of Quotients, Polynomial Rings: Notation and Terminology, The Division Algorithm and Consequences.