– Unit-I – 1.1Historical background : 1.1.1A brief historical background of the Algebra in the context of India and Indian heritage and culture 1.1.2A brief biography of Brahmagupta 1.2Groups, Subgroups and their basic properties 1.3Cyclic groups 1.4Coset decomposition 1.5Lagrange’s and Fermat’s theorem 1.6Normal subgroups 1.7Quotient groups – Unit-II – 2.1Homomorphism and Isomorphism of groups 2.2Fundamental theorem of homomorphism 2.3Transformation and Permutation group Sn(n < 5) 2.4Cayley’s theorem 2.5Group automorphism 2.6Inner automorphism 2.7Group of automorphisms – Unit-III – 3.1Definition and basic properties of rings 3.2Ring homomorphism 3.3Subring 3.4Ideals 3.5Quotient ring 3.6Polynomial ring 3.7Integral domain 3.8Field – Unit-IV – 4.1Definition and examples of Vector space 4.2Subspaces 4.3Sum and direct sum of subspaces 4.4Linear span, Linear dependence, Linear independence and Their basic properties 4.5Basis 4.6Finite dimensional vector space and dimension 4.6.1Existence theorem 4.6.2 Extension theorem 4.6.3Invariance of the number of elements 4.7Dimension of sum of subspaces 4.8Quotient space and its dimension – Unit-V – 5.1Linear transformation and its representation as a matrix 5.2Algebra of linear transformation 5.3Rank-Nullity theorem 5.4Change of basis, dual space, bi-dual space and natural isomorphism 5.5Adjoint of a linear transformation 5.6Eigenvalues and Eigenvectors of a linear transformation 5.7Diagonalization