These notes are the basis for our Math 4580 lectures. There may be material discussed in class which is not included in these notes, and vice versa.
Lecture 1: Intro, set theory review
Lecture 2: More review of set theory and functions; equivalence relations
Lecture 3: Equivalence relations and partitions
Lecture 4: Induction, well-ordering, division algorithm
Lecture 5: Greatest common divisors, Euclidean algorithm, primes
Lecture 6: Binary operations
Lecture 7: Examples of groups
Lecture 8: More examples, basic properties of groups
Lecture 9: Subgroups
Lecture 10: Cyclic subgroups
Lecture 11: Cyclic groups
Lecture 12: Classification of cyclic groups, the order of ak
Lecture 13: Subgroups of cyclic groups
Lecture 14: Symmetric groups
Lecture 15: Cycle notation
Lecture 16: Transpositions, even and odd permutations, dihedral groups
Lecture 17: Relations in Dn
Lecture 18: Generators and relations
Lecture 19: Proof that the sign of a permutation is well-defined
Lecture 20: Alternating groups
Lecture 21: Cosets
Lecture 22: Lagrange's theorem
Lecture 23: Homomorphisms and isomorphisms
Lecture 24: More about homomorphisms
Lecture 25: Structural properties
Lecture 26: Cayley's theorem, direct products
Lecture 27: Products of cyclic groups
Lecture 28: External vs. internal direct products, normal subgroups
Lecture 29: Normal subgroups, coset multiplication
Lecture 30: Quotient groups
Lecture 31: Kernels
Lecture 32: First Isomorphism Theorem
Lecture 33: Simple groups
Lecture 34: Rings, zero divisors and units
Lecture 35: Cancellation, integral divisors, division rings, fields
Lecture 36: Subrings, ring homomorphisms
Lecture 37: Ideals, the characteristic of a ring
Lecture 38: Coset multiplication, quotient rings
Lecture 39: First isomorphism theorem for rings