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Contents

• V Full version
• 0 Introduction
• 1 Groups and homomorphisms
• 1.1 Groups
• 1.2 Homomorphisms
• 1.3 Cyclic groups
• 1.4 Dihedral groups
• 1.5 Direct products of groups
• 2 Symmetric group I
• 2.1 Symmetric groups
• 2.2 Sign of permutations
• 3 Lagrange's Theorem
• 3.1 Small groups
• 3.2 Left and right cosets
• 4 Quotient groups
• 4.1 Normal subgroups
• 4.2 Quotient groups
• 4.3 The Isomorphism Theorem
• 5 Group actions
• 5.1 Group acting on sets
• 5.2 Orbits and Stabilizers
• 5.3 Important actions
• 5.4 Applications
• 6 Symmetric groups II
• 6.1 Conjugacy classes in Sn
• 6.2 Conjugacy classes in An
• 7 Quaternions
• 8 Matrix groups
• 8.1 General and special linear groups
• 8.2 Actions of GLn(C)
• 8.3 Orthogonal groups
• 8.4 Rotations and reflections in R2 and R3
• 8.5 Unitary groups
• 9 More on regular polyhedra
• 9.1 Symmetries of the cube
• 9.2 Symmetries of the tetrahedron
• 10 Mobius group
• 10.1 Mobius maps
• 10.2 Fixed points of Mobius maps
• 10.3 Permutation properties of Mobius maps
• 10.4 Cross-ratios
• 11 Projective line (non-examinable)