0. Outline for the Notes
What are we trying to do here?
This series will be focused on basic Group Theory, but with a slight twist. Semigroups will also be introduced as a way to give intuition about certain ideas in Group Theory, which is not common in Algebra textbooks. It is not intended to be a course on Semigroup Theory—or even Group Theory—but rather as a companion of sorts to a first course in Group Theory. Recommended reading for such a first course would be A First Course in Abstract Algebra by Fraleigh, which is commonly used, but is far from the only option.
Below is an outline of topics:
preliminaries: a reminder of naïve set theory, and definitions of relations, functions, and the cartesian product
generators and Cayley graphs
Green’s relations, and the ax=b & ya=b definition of a group
monogenic semigroups and cyclic groups
homomorphisms, congruences, and isomorphisms
the First Isomorphism Theorem
transformation semigroups and symmetric groups
actions on a set
important theorems: Cayley, Lagrange, Cauchy
free objects and presentations
The outline may be changed during development to avoid circular logic and for the convenience of the reader, as I am not directly mimicking any particular textbook.