Level 7.2 – Groups

These notes are based on lectures by Henry Wilton in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.

Summary of the course [Expand]

• Lectures 1-3 are about the motivation and definition of a group, and very basic examples and properties, including definitions of symmetric groups and subgroups.
• Lectures 4-5 are about isometries and dihedral groups.
• Lectures 6-7 are about homomorphisms and isomorphisms and their basic properties. We also introduce cyclic groups.
• Lecture 8 is about cosets and Lagrange's theorem and its proof.
• Lectures 9-11 introduce group actions and the orbit-stabiliser theorem, cauchy's theorem, and the fact that every group is isomorphic to a subgroup of a symmetric group.
• Lectures 12-13 are about the mobius group and its basic properties.
• Lectures 14-15 are about classifying all the groups of size up to 8.
• Lectures 16-17 are about normal subgroups, quotient groups and the first isomorphism theorem.
• Lectures 17-20 are about permutations and alternating groups, and the proof A5 is simple.
• Lectures 20-22 are about matrix groups, orthogonal groups, and their basic properties.
• Lectures 23-24 are about finding the symmetry groups of platonic solids.

See my groups notes here