- August 28: Introduction. Groups. Associativity as a consequence of
function composition. Examples of groups. Symmetric groups. Symmetries.
*GL*(_{n}**R**). - August 30: Dihedral groups. Generators for the dihedral groups. Normal form for the dihedral group. Group presentations (somewhat informal). Cyclic groups. Order of a group. Order of an element. Homework #1
- September 1: Homomorphisms. Isomorphisms. Heisenberg group. Subgroups.
- September 4:
*Labor Day* - September 6: Centralizers and center. Normalizers. Stabilizers. Conjugation. Cyclic permutations. Cyclic decomposition. Homework #2
- September 8: Direct sums and products. Generating subgroups by subsets.
- September 11: Uniqueness of cyclic groups. Posets.
- September 13: Zorn's lemma. Axiom of choice. Maximal subgroups in finitely generated groups. Homework #3
- September 16: Proof of Zorn's lemma. Notes on the Axiom of Choice.
- September 18: Normal subgroups. Quotient groups. Kernels. Cosets. Index.
- September 20: Index. Examples of quotients. First and third isomorphism theorems.
- September 22: Normality is not transitive. Index-2 subgroups are normal. Cardinality of the product of subgroups. Simple groups. Jordan–Hölder theorem (statement). Homework #4
- September 25: Sign of a permutation. Alternating group. Alternating group is generated by 3-cycles. 15 puzzle.
- September 27: Alternating group is simple. Permutation representations. Linear representations. Orbits. Homework #5
- September 29:
**Test** - October 2: Subgroups of index p. Stabilizers. Sizes of orbits in general, and conjugacy classes in particular. (Finite)
*p*-groups. Groups of order*p*are abelian. Right group actions.^{2} - October 4: Inner and outer automorphisms. Aut(ℤ/
*n*ℤ). Statement of Sylow theorems. Cauchy's theorem for abelian groups. Homework #6 - October 6: First Sylow's theorem.
- October 9: Second and third Sylow's theorems. Groups of order
*pq*. Recognizing direct products. - October 11: Semidirect products. Rigid motions of
**R**^{n}. Homework #7 - October 13: Words. Group words. Free groups. Universal property of a free group.
- October 16: Reduced words. Presentations.
- October 18: Rings. Examples of rings.
*M*(_{n}*R*). Units. Zero divisors. Polynomial rings. Homework #8 - October 20:
*Midsemester break* - October 23: Polynomial rings. Power series. Laurent series. Group rings. Ring homomorphisms. Ideals.
- October 25: Quotient rings. Evaluation homomorphism. Polynomials vs polynomial maps. Operations on ideals. Homework #9
- October 27: Ideals generated by subsets. Ideals, rings, fields generated by a set. Complex numbers as a quotient ring. Prime ideals. Integral domains.
- October 30: Categories. Examples of categories. How to stop worrying and love the universes. Products. Uniqueness of products. Opposite category. Coproducts. Coproducts.
- November 1: Products and coproducts in the category of abelian groups. Products and coproducts in the category of groups (without proofs). Free products. Fundamental groups (without proofs). Rings of fractions (localization). Homework #10
- November 3: Universal property of localization. Chinese remainder theorem. Chinese remainder theorem and Lagrange interpolation formula.
- November 6:
**Test** - November 8: Euclidean domains. Gaussian integers. Principal ideal domains. Prime elements and irreducible elements (part I). Homework #11
- November 10:
*No class* - November 13: Prime elements and irreducible elements (part II). Divisibility. Associates.
*Guest lecture by James Cummings* - November 15: Unique factorization domains. Noetherian rings. Noetherian induction.
*Guest lecture by James Cummings* - November 17: PIDs are UFDs. Greatest common divisor. Polynomial rings over UFDs (part I).
*Guest lecture by James Cummings* - November 20: Polynomial rings over UFDs (part II). Irreducibility criteria (part I). Homework #12
- November 27: Irreducibility criteria (part II). Cyclotomic polynomials of prime order. Polynomial ideals as consequences of systems of polynomial equations. Hilbert's basis theorem.
- November 29: Monomial orderings. The division algorithm. Gröbner bases. Monomial ideals.
- December 1: Syzigies.
- December 4: Buchberger's criterion. Buchberger's algorithm.
- December 6: Gröbner bases can be huge. Minimal Gröbner bases. Reduced Gröbner bases. Uniqueness of reduced Gröbner bases.
- December 8: Elimination ideals. Computing elimination ideals via Gröbner bases.
- December 11–17:
**Take-home final exam.**Final exam rules