Week	Brief Description
1	Review of groups, rings and fields. Substructures: subgroups, subrings and subfields. Structure preserving maps: homomorphisms and isomorphisms.
2	Group homomorphisms, kernel, cosets. Lagrange theorem, normal subgroup. Isomorphism theorems, composition series, Holder's program. Ring homomorphisms, ideals, kernels factor rings. Direct products.
3	Symmetric groups, conjugacy classes, groups acting on sets and Polya's enumeration theorem.
4	Class equation and the Sylow theorems.
5	Vector spaces, subspaces, linear transformations, bases, direct sums. Inner product spaces, normed spaces, orthogonal sets.
6	Gram-Schmidt process. Matrices, row-operations, rank. Matrices associated to linear transformations.
7	Change of basis, determinants. Polynomials, unique factorization
8	Cayley-Hamilton Theorem. Normal forms, characteristic equation, Smith normal form, Jordan canonical form, rational canonical form
9	Introduction to modules