- Groups (5 weeks)
- Direct products, Normal subgroups, Quotient groups, and the
- Groups acting on sets; orbits and stabilizers. Applications:
class formula, centralizers and normalizers, centers of finite
p-groups. Conjugacy classes of Sn
- Sylow's Theorems, Solvable groups, Simple groups, simplicity
of An. Examples: Finite groups of small order (8).
- Structure of finitely generated abelian groups. Free
References: Lang, Chapter I; Dummit and Foote, Part I; Rotman.
- Basic linear algebra (3 weeks)
- Vector spaces, Linear dependence/independence, Bases, Matrices
and linear maps. Dual vector space, quotient vector spaces,
- Determinants, basic properties. Eigenspaces and eigenvectors,
- Inner products and orthonormal sets. Spectral theorem for
normal operators (finite dimensional case).
References: Lang, Chapters XIII and XIV; Dummit and Foote, Chapter 11.
- Rings, modules and algebras (6 weeks)
- Rings, subrings, fields, ideals, homomorphisms, isomorphism
theorems, polynomial rings.
- Integral domains, Euclidean domains, PID's. UFD's and Gauss's
F[x1,..., xn] is an UFD). Examples.
- Prime ideals, maximal ideals. The Chinese remainder
Theorem. Fields of fractions.
- The Wedderburn Theorem (no proof). Simplicity and semisimplicity.
- Noetherian rings and the Hilbert Basis Theorem.
- Finitely generated modules over PID's, the structure theorem.
References: Lang, Chapters II, III, V, and VI; Jacobson, Chapter 2;
Dummit and Foote, Part II.