**Tuesdays 10 AM to 12 PM**

- Week 1 (May 18):
Preliminary Review
- Subgroups, quotients, isomorphism theorems, cosets, index of a subgroup
- One step subgroup test

- Cauchy, Lagrange
- Everything about: cyclic, symmetric, alternating, dihedral groups of order ≤ 20
- Especially useful problems: if G is nonabelian of order p^3, then Z(G) = [G, G])

- Subgroups, quotients, isomorphism theorems, cosets, index of a subgroup
- Week 2 (May 25):
Finite Groups
- Special types of groups, the symmetric group, p-groups
- Series of groups, solvable, simple, nilpotent; Jordan-Holder theorem
- Group actions, orbit-stabilizer, class equation,
- Cayley representation, permutation representation
- FTFGAG: The Fundamental Theorem of Finitely Generated Abelian Groups
- Invariant factors
- Elementary divisors
- How to go back and forth

- Recognition of direct products and semidirect products

- Week 3 (June 1):
Sylow Theorems
- Showing groups are abelian
- Classification: isomorphism classes of groups of a given order, recognizing direct and semidirect products

- Week 4 (June 8):
Rings and Commutative Algebra
- Morphisms, Ideals, quotients, zero divisors, isomorphism theorems, CRT
- Irreducible and prime elements, nilpotent, units
- Radical, nilradical, spec and maxspec
- Special types: domains, integral domains, Euclidean ⇒ PID ⇒ UFD ⇒?, Dedekind domains, Noetherian, Artinian
- Zorn's lemma arguments
- Bonus optional stuff: localization

- Week 5 (June 15):
Modules and Homological Algebra
- Morphisms, submodules, isomorphism theorems, principal ideals
- Free and projective, free rank, torsion submodule, annihilators
- Tensor product
- SESs and splitting