Prof. Ilya Tyomkin
Time and Place:
- Groups as symmetries. Examples: cyclic, dihedral, symmetric, matrix groups.
- Homomorphism. Subgroups and normal subgroups. Quotient groups. Lagrange’s theorem. The isomorphism theorems. Direct products of groups.
- Actions of groups on sets. Cayley’s theorem.
- Group automorphisms.
- Sylow’s theorems. Application: classification of groups of small order.
- Composition series and Jordan–Hoelder theorem. Solvable groups.
- Classification of finite abelian groups, finitely-generated abelian groups.
- Symmetric group and alternating group. The alternating group is simple.
- Rings, maximal and prime ideals, integral domain, quotient ring. Homomorphism theorems.
- Multilinear algebra: Quotient spaces. Tensor products of vector spaces. Action of $S_n$ on tensor powers. Exterior and symmetric algebras. Multilinear forms and determinant.
- Optional topics: group of symmetries of platonic solids, free groups, semidirect products, representation theory of finite groups.