Math 511 - Fall 2002

Week one:
  • 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.

    Week two:
  • Monday (Sep 9): Review of number theory and linear algebra (continue), invariant subspaces.
  • Wednesday: Definition of group.
  • Friday: Subgroups, homomorphisms.

    Week three:
  • Monday (Sep 16): Finite permutation groups.
  • Wednesday: Finite permutation groups, direct product, direct sum.
  • Friday: Cosets, normal subgroup, quotient group, kernel.

    Week four:
  • Monday (Sep 23): Quotient group, kernel.
  • Wednesday: 1st, 2nd, 3rd isomorphism theorems, problem session.
  • Friday: -

    Week five:
  • Monday (Sep 30): Definition of ring.
  • Wednesday: Ideals, kernel, quotient ring.
  • Friday: Isomorphism theorems.

    Week six:
  • Monday (Oct 7): Chinese remainder theorem, maximal, prime ideals.
  • Wednesday: Polynomial rings.
  • Friday: Polynomial rings (continue).

    Week seven:
  • Monday (Oct 14): Holiday
  • Wednesday: Unique factorization.
  • Friday: Principal ideal domains.

    Week eight:
  • Monday (Oct 21): Principal ideal domains (continue).
  • Wednesday: Euclidean domains, rings of fractions.
  • Friday: Localization, group actions.

    Week nine:
  • Monday (Oct 28): Group actions, equivalence, orbit stabilizer theorem.
  • Wednesday: Orbit stabilizer theorem (continue), examples.
  • Friday: More examples.

    Week ten:
  • Monday (Nov 4): The Sylow theorems.
  • Wednesday: The Sylow theorems (continue).
  • Friday: -

    Week eleven:
  • 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, ...

    Week twelve:
  • 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.

    Week thirteen:
  • Monday (Nov 25): -
  • Wednesday: Cyclic modules, finitely generated modules, direct sums.
  • Friday: Free modules, homomorphisms, matrices, opposite ring.

    Week fourteen:
  • Monday (Dec 2): Free modules over a PID.
  • Wednesday: Classification of finitely generated modules over a PID.
  • Friday: Application to finitely generated abelian groups.