NOTE: Attempt any FIVE questions selecting at least TWO questions from each section.

Section-I (5/9)

Groups

•    Definition and examples of groups

•    Subgroups lattice, Lagrange’s theorem

•    Cyclic groups

•    Groups and symmetries, Cayley’s theorem

Complexes in Groups

•    Complexes and coset  decomposition of groups

•    Centre of a group

•    Normalizer in a group

•    Centralizer in a group

•    Conjugacy classes and congruence relation in a group

Normal Subgroups

•    Normal subgroups

•    Proper and improper normal subgroups

•    Factor groups

•    Isomorphism theorems

•    Automorphism group of a group

•    Commutator subgroups of a group

Permutation Groups   

•    Symmetric or permutation group

•    Transpositions

•    Generators of the symmetric and alternating group

•    Cyclic permutations and orbits, The alternating group

•    Generators of the symmetric and alternating groups

Sylow Theorems

•    Double cosets

•    Cauchy’s theorem for Abelian and non-Abelian group

•    Sylow theorems (with proofs)

•    Applications of Sylow theory

•    Classification of groups with at most 7 elements



Section-II (4/9)

Ring Theory

•    Definition and examples of rings

•    Special classes of rings

•    Fields

•    Ideals and quotient rings

•    Ring Homomorphisms

•    Prime and maximal ideals

•    Field of quotients

Linear Algebra

•    Vector spaces, Subspaces

•    Linear combinations, Linearly independent vectors

•    Spanning set

•    Bases and dimension of a vector space

•    Homomorphism of vector spaces

•    Quotient spaces

Linear Mappings

•    Mappings, Linear mappings

•    Rank and nullity

•    Linear mappings and system of linear equations

•    Algebra of linear operators

•    Space L( X, Y) of all linear transformations

Matrices and Linear Operators

•    Matrix representation of a linear operator

•    Change of basis

•    Similar matrices

•    Matrix and linear transformations

•    Orthogonal matrices and orthogonal transformations

•    Orthonormal basis and Gram Schmidt process

Eigen Values and Eigen Vectors

•    Polynomials of matrices and linear operators

•    Characteristic polynomial

•    Diagonalization of matrices