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.