Graduate Abstract Algebra I

Rutgers MATH 551, Fall 2019.

daniel.krashen+alg1@gmail.com

Lecture 1: Introduction to algebraic structures via universal algebra

Lecture 2: Monoid-oids (aka Categories), some general remarks about groups

Lecture 3: The orbit-stabilizer theorem and the class equation

Lecture 4: The class equation!

Lecture 5: Putting groups together from their pieces

Lecture 6: Putting groups back together -- group extensions

Lecture 7: An accounting of structures, Nilpotent groups, part 1

Lecture 8: Free groups, part 1

Lecture 9: Free groups, part 2: universal properties and adjunctions

Lecture 10: Nilpotent groups, part 2; Solvable groups

Lecture 11: Rings: definitions and examples

Lecture 12: Basic properties of rings and ideals; fractions and localization, part 1

Lecture 13: Fractions and localization, part 1; Chinese remainder theorem; Principal ideal domains

Lecture 14: Unique factorization domains and Gauss' Lemma

Lecture 15: The Hilbert basis theorem; Introduction to modules

Lecture 16: More modules; tensor products, part 1

Lecture 17: Tensor products, part 2

Lecture 18: Tensor products, part 3; bilinear forms; intro to vector spaces

Lecture 19: Vector space notions: duals, matrices, Kroneker products, tensor algebras

Lecture 20: Symmetric and Exterior algebras. Start of modules over a PID

Lecture 21: The structure of modules over a PID

Lecture 22: Examples and application of the structure theory for modules over a PID

Lecture 23: Jordan-Holder for X-groups. Setting the stage for the Yoneda Lemma

Lecture 24: Some categorical notions and the Yoneda Lemma. Introduction to representations of finite groups.

Lecture 25: Representations of finite groups, part 2.

Lecture 26: Representations of finite groups, part 3; Burnside's Theorem.