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 24 & Aug 26 |
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. |
|
| Aug 31 & Sep 2 |
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 7 & Sep 9 |
Action by conjugation and class equation. Solvability of
p-groups. Sylow theorems.
Ch.4, §§4.3-4.5. |
|
| Sep 14 & Sep 16 |
Sylow theorems and their applications. Classification of groups of small
orders. Introduction to symmetric and alternating groups. Ch.4, §§4.5-4.6 and Ch.5, §§5.3-5.5. |
|
| Sep 21 & Sep 23 |
Symmetric and alternating
groups continued. 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. |
|