1. General linear groups, Permutation groups & representation theory

2. General linear group: GL(n, R/C) GL(n, R/C) denotes the set of all n x n invertible matrices with real/complex coefficients. Operation: matrix multiplication Identity element: the identity matrix Inverse of an element: matrix inverse Not commutative (not Abelian)

3. More on GL(n, R/C) GL(n, R/C) has two disconnected subgroups – GL+(n, R/C), GL-(n, R/C) Each element of GL(n, R) is a linear map. – They do not commute. – The matrices of rotations make a sub-group

4. Permutation group: is the set of all permutations on n distinct elements. Binary operation: composition

7. Representation Theory Representation theory studies how any given abstract group can be realized as a group of matrices.

11. “quality” of the representation http://en.wikipedia.org/wiki/Group_representation If G is a topological group and V is a topological vector space,topological vector space a continuous representation of G on V is a representation ρ such that the application Φ : G × V → V defined by Φ(g, v) = ρ(g)(v) is continuous.continuous Faithful Isomorphism