Foundations of Group Theory

Define a group and verify the group axioms for physical examples such as rotations, permutations, and matrix groups.

Identify subgroups of a given group and apply Lagrange's theorem.

Define conjugacy classes and compute them for a finite group.

Construct cosets of a subgroup and determine whether a subgroup is normal.

Construct quotient groups from normal subgroups and explain their structure.

Determine whether a map between groups is a homomorphism or isomorphism.

Construct direct product groups and identify them in physical systems with multiple independent symmetries.