Abstract Algebra

Unit 1 · Isomorphism Theorems, Cyclic & Permutation Groups

Group homomorphism and isomorphism with properties; first, second and third isomorphism theorems for groups; cyclic groups and their properties, classification of subgroups of cyclic groups; permutation groups and their properties, even and odd permutations, Cayley's theorem.

Unit 2 · Ring, Field & Integral Domain, Ideals

Definition and properties of a ring, examples, subrings, integral domains and fields, characteristic of a ring and field, ring homomorphism, ideals and quotient rings, field of quotients of an integral domain, Euclidean rings, polynomial rings, polynomials over the rational field, Eisenstein criterion, polynomial rings over commutative rings, and unique factorization domains.

Unit 3 · Vector Spaces

Definition and examples of vector spaces, subspaces, sum and direct sum of subspaces, linear span, linear dependence and independence and their basic properties, basis, finite-dimensional vector spaces, existence theorem for bases, invariance of the number of elements of a basis, dimension, existence of a complementary subspace, dimension of sums of subspaces, quotient space and its dimension.

Unit 4 · Linear Transformation

Linear transformations and their representation as matrices, the algebra of linear transformations, the rank-nullity theorem, change of basis, dual space, bi-dual space and natural isomorphism, and the adjoint of a linear transformation.