Tutorial 1 (Substituted by Momchil Konstantinov)

• Groups (Definition and examples), 

• Symbols and Relations : Klein four group, 

• Isomorphism via multiplication table

Tutorial 2 (Substituted by Momchil Konstantinov)

• Homomorphism : kernel and image, 

• Reminders on subgroups and cyclic groups, 

• Left (and right) cosets : Lagrange Theorem, 

Tutorial 3 

• More on cosets, 

• Normal subgroups : Left cosets = Right cosets,

• Normal criterion I : conjugation 

Tutorial 4 

• Normal criterion II : Kernel is normal,  

• The first isomorphism theorem, 

• Conjugacy classes I : Normal subgroups as a union of conjugacy classes

Tutorial 5 

• Conjugacy classes II : Centre of a group

• More on the first isomorphism theorem,

• Technique in choosing your homomorphism (as opposed to saying `obviously...')

Tutorial 6

• Automorphisms : inner & outer 

• Proof that G/Z(G) is isomorphic to I(G)

• Cheap proof that Z(GL(2,K))=scalar matrices

Tutorial 7

• Standard matrix groups: O(N), U(N) and their S-variants, 

• Cheap proof : center of U(N) (for N>1) ,

• Common misconceptions : center and normality revisit

Tutorial 8

• Semi-direct product : definitions, 

• Examples : O(N)=Z_2 ⋉ O(N) and D_n=Z_2 ⋉ Z_n, 

• Worked Example : U(N)=U(1) ⋉ SU(N) 

Tutorial 9

• Euclidean group I : isometries in Euclildean space, 

• SU(2) as 3-sphere

•  Pauli matrices and their (quanternion-like) properties

Tutorial 10

• Review : orthogonal transformations and dot product, 

• O(N)-vectors and O(N)-scalars : definitions & examples, 

• Euclidean group II : Invariance of laws .