ABSTRACT
We conclude the textbook with a chapter on group theory topics. The first three sections introduce group actions, along with orbits, stabilizers, the Cauchy-Frobenius Lemma, and blocks in transitive group actions. Applying these techniques to the action of a group on itself by left multiplication and the action of conjugation, we revisit Cayley's Theorem and prove the Class Equation. Still using techniques from group actions, we also prove Sylow's Theorem, a deep theorem on the internal structure of finite groups. After a section that introduces the semidirect product of two groups, we use Sylow's Theorem to explore classification problems of groups by their order.
