Department of Mathematics
Stony Brook University
office: Math Tower 5-111
phone: (631) 632-8287
e-mail: leon.takhtajan@stonybrook.edu
| Dates | Sections
covered and assigned reading |
Homework |
| Aug 27 & Aug 29 |
Definition of a
group. Examples: symmetric group, cyclic group, dihedral group,
other groups of symmetry. Homomorphisms and
isomorphisms. Subgroups.
Order of an element and cyclic subgroups. Cosets and Lagrange's theorem. Order of an element divides the order of the group. Normal subgroups and quotient groups.
Chapters 1-2 and §§3.1-3.2 in Chapter 3. |
|
| Sep 3 & Sep 5 |
Quotient groups. Isomorphism theorem. Direct and semidirect products. Simple groups.
Holder's theorem (no proof). Group actions on sets; orbits and stabilizers.
Ch.3, §§3.3-3.5, Ch.4, § 4.1 and Ch. 5, §5.1 and §§5.4-5.5 (up to p. 180). |
|
| Sep 10 & Sep 12 |
Action by conjugation and class equation. Solvability of
p-groups. Sylow theorems.
Ch.4, §§4.3-4.5. |
|
| Sep 17 & Sep 19 |
Application of Sylow theorems. Classification of groups of small
orders. Symmetric and alternating groups. Ch.4, §§4.5-4.6 and Ch.5, §§5.3-5.5. |
|
| Sep 24 & Sep 26 |
Classification of finitely
generated abelian groups, main theorem and unqieness. Torsion
subgroup and rank. Free groups.
Ch.5, §5.2, Ch.6, §6.1 pp. 196-198, §6.3. §§5.3-5.5, and S. Lang Algebra, Ch. 1, §8. |
|
| Oct 1 & Oct 3 |
Rings. Definitions,
examples. Quaternions and ℤ[d]. Subrigns, homomorphisms.
Midterm I on Oct 3.
Ch 7, §§7.1-7.3. |
|
| Oct 8 & Oct 10 |
Ideals and quotients. Theorem: Z in a principal ideal domain. Maximal ideals.
Integral domains and PIDs. Field of fractions. Ideals in integral
domains. Relation between operations with ideals (sum, product,
intersection) and and operations with elements (product, lcm, gcd) -
for a PID. Euclidean domains and U.F.D.
Ch 7, §§7.4-7.6 & Ch 8, §§8.1-8.3. |
|
| Oct 17 |
Arithmetics of the field of Gaussian integers. Chinese remainder
theorem. Polynomial rings. Irreducible polynomilas. Roots and divisibility.
Ch 8, §8.3 & Ch 9, §§9.1-9.3. |
|
| Oct 22 & Oct 24 |
Gauss Lemma and Eisenstein's criterion.
Polynomials in several variables. Unique factorization. Noetherian rings
and Hilbert basis theorem. Modules.
Ch 9, §§9.3-9.5 (up to p. 317) & Ch. 10, §§10.1-10.2. |
|
| Oct 29 & Oct 31 |
Structure theorem for modules over a PID. Vector spaces and linear operators: basic theory. Basis and dimension.
Ch 12, §12.1 & Ch. 11, §§11.1-11.3. |
All late homework is due. |
| Nov 12 & Nov 14 |
Tensor product of vector spaces, tensor, symmetric and exterior
algebras. Determinants.
Ch. 11, §§11.4-11.5. |
|
| Nov 19 & Nov 21 |
Eigenvalues and diagonalization. Rational canonical form and Jordan
normal form.
Ch. 12, §§12.2-12.3. |
|
| Nov 26 |
Hermitian and Euclidean inner products, orthonormal sets in
finite-dimensional vector spaces. Schur decomposition theorem.
Lang, Ch. XV, §§3-5 and notes from the class. |
|
| Dec 1 & Dec 5 |
Spectral theorem for unitary, Hermitian and normal operators. Review.
Lang, Ch. XV, §§ 5-7 and notes from the class. |
|