Plan for the lectures

Part I - Groups

• Monday November 2, sections 1.1-1.3
  • Groups - basic axioms and examples
  • Dihedral groups
  • Symmetric groups
• Monday November 9, sections 1.3-1.7
  • Matrix groups
  • The quaternion group
  • Homomorphisms and isomorphisms
  • Group actions
• Monday November 16, sections 2.1-2.4
  • Subgroups - definitions and examples
  • Centralizers and normalizers, stabilizers and kernels
  • Cyclic groups and cyclic subgroups
  • Subgroups generated by a subset of a group
• Monday November 23, sections 3.1-3.3
  • Quotient groups and homomorphisms - definitions and examples
  • More on cosets and Lagrange's Theorem
  • The Isomorphism Theorems
• Monday November 30, sections 4.1.4-3
  • Group actions and permutation representations
  • Groups acting on themselves by left multiplication and Cayley's Theorem
  • Groups acting on themselves by conjugation and the Class Equation
• Monday December 7, sections 4.4-4.5
  • Automorphisms
  • The Sylow Theorems
• Monday December 14, sections 5.1-5.2
  
  • Direct products
  • The Fundamental Theorem of finitely generated abelian groups

Part II - Rings

• Monday January 25, sections 7.1-7.2
  • Rings, fields, integral domain, examples
• Monday February 1, sections 7.3-7.4
  • Homomorphism, kernel, ideal
  • Operations on ideals, prime ideal
• Monday February 8, sections 7.5-7.6
  • Rings of fractions
  • Chinese Remainder Theorem
• Monday February 15, sections 8.1-8.2
  • Euclidean domains
  • P.I.D.
• Monday February 22, section 8.3
  • U.F.D.
  • Quadratic integer rings
• Monday February 29, sections 9.1-9.4
  • Polynomial rings
  • Irreducibility criteria
• Monday March 7, sections 10.1-10.3
  • Modules, vector spaces