Groups as symmetries. Examples: cyclic, dihedral, symmetric and 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.