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.