Math 511 - Fall 2002
Monday (Sep 2): Review of number theory and linear algebra.
Greatest common divisor of two integers and the Euclidean algorithm.
Greatest common divisor of two polynomials.
Wednesday : Review of number theory and linear algebra (continue).
Finite field with 8 elements, eigenvectors and eigenvalues.
Friday: Review of number theory and linear algebra (continue), minimal polynomial, characteristic polynomial, diagonalizability and Jordan form.
Monday (Sep 9): Review of number theory and linear algebra (continue), invariant subspaces.
Wednesday: Definition of group.
Friday: Subgroups, homomorphisms.
Monday (Sep 16): Finite permutation groups.
Wednesday: Finite permutation groups, direct product, direct sum.
Friday: Cosets, normal subgroup, quotient group, kernel.
Monday (Sep 23): Quotient group, kernel.
Wednesday: 1st, 2nd, 3rd isomorphism theorems, problem session.
Monday (Sep 30): Definition of ring.
Wednesday: Ideals, kernel, quotient ring.
Friday: Isomorphism theorems.
Monday (Oct 7): Chinese remainder theorem, maximal, prime ideals.
Wednesday: Polynomial rings.
Friday: Polynomial rings (continue).
Monday (Oct 14): Holiday
Wednesday: Unique factorization.
Friday: Principal ideal domains.
Monday (Oct 21): Principal ideal domains (continue).
Wednesday: Euclidean domains, rings of fractions.
Friday: Localization, group actions.
Monday (Oct 28): Group actions, equivalence, orbit stabilizer theorem.
Wednesday: Orbit stabilizer theorem (continue), examples.
Friday: More examples.
Monday (Nov 4): The Sylow theorems.
Wednesday: The Sylow theorems (continue).
Monday (Nov 11): p-groups, composition series.
Wednesday: Jordan-Holder theorem, simplicity of the dodecahedral group.
Friday: Isomorphism of the dodecahedral group and A5, composition series of S5, simplicity of A6, A7, ...
Monday (Nov 18): Solvable and nilpotent groups.
Wednesday: Solvable and nilpotent groups (continue).
Friday: Modules basics: definitions, examples, left, right modules, submodules, quotient modules, isomorphism theorems.
Monday (Nov 25): -
Wednesday: Cyclic modules, finitely generated modules, direct sums.
Friday: Free modules, homomorphisms, matrices, opposite ring.
Monday (Dec 2): Free modules over a PID.
Wednesday: Classification of finitely generated modules over a PID.
Friday: Application to finitely generated abelian groups.