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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
Part II - Rings
- Monday December 14, sections 5.1-5.2
- Direct products
- The Fundamental Theorem of finitely generated abelian groups
- Monday January 25, sections 7.1-7.2
- Rings, fields, integral domain, examples
- Monday February 1, sections 7.3-7.4
- Homomorphisms, kernel, ideal
- Operations on ideals, prime ideal