Part 1: Group theory Definition of a group Groups as symmetries Examples: cyclic, dihedral, symmetric, matrix groups Homomorphisms Subgroups and quotient groups Cosets Conjugacy classes Normal subgroups Lagrange's theorem The isomorphism theorems Actions of groups on sets Symmetric group and alternating group Cayley's theorem Groups of symmetries of plactonic solids Direct products of groups Group automorphisms Sylow's theorems Applications: classification of groups of small order The alternating group is simple Classification of finite abelian groups, finitely-generated abelian groups Composition series Jordan-Hoelder theorem Nilpotent and solvable groups Free groups Part 2: Ring theory Definition and examples Ring homomorphisms Ideals Chinese Remainder Theorem Ideals, prime ideals Varieties Principal ideal domains Monomial ideals Grobner basis and Hilbert basis theorem Radical Ideals, Hilbert Nullstellensatz Varieties and dimension.