Short Course Description
Prerequisites
Linear algebra 2A.
Course description.
The course is an introduction to the classical group theory. We will describe fundamental objects and concepts of the theory and provide examples. We will mostly deal with finite groups, but infinite groups will show up as well.
Outline of the course
1. Definitions and examples. Commutative and noncommutative groups. Cyclic groups. Homomorphisms, kernels and images of homomorphisms. Center of a group.
2. Symmetric groups, cycles and transpositions. Decomposition of a permutation into cycles. Systems of generators. Sign of a permutation, alternating subgroups.
3. Multiplicative groups of a field, cyclicity of its subgroups.
4. Left and right cosets, Lagrange theorem. Normal subgroups, quotient groups. Quotient of a group by the kernel of a homomorphism.
5. Conjugation, conjugacy classes, the permutation group case. The group of automorphisms of a group, internal automorphisms.
6. Commutators of elements, commutator subgroups. Basic properties of the commutator subgroups. Solvable and nilpotent groups.
7. Actions of groups on sets. Transitive, free and faithful actions. Orbits and stabilizers. Natural projection from a group to an orbit.
8. Left and right actions of a group on itself. Conjugation action and a homomorphism from a group to its automorphism group.
9. Products of groups and semi-direct products. Semi-direct products of cyclic groups and group automorphisms.
10. Orthogonal groups in three-dimensional spaces, proper and nonproper transformations, groups of symmetries of platonic solids.
11. Simple groups, p-groups, Sylow theorems, classification of groups of small order.
12. Free groups with finite number of generators, universal property. Generators and relations presentation, example of dihedral groups.
13. Finitely generated abelian groups, the fundamental theorem.
14. Time permitting: action of symmetric groups on polynomials. Symmetric polynomials.
Assignments
Weekly homework assignments.
Grading
weekly assignments -- 10 % of grade (of the best 10 homeworks)
final exam -- 90% of the grade
Bibliography
D. S. Dummit and R. M. Foote, Abstract algebra, second edition
J. F. Humphreys, A course in group theory.
M.Artin, Algebra, second edition.
Full syllabus is to be published
