• Modules: definitions, first examples; basic properties; submodules; factor modules; isomorphism theorems; correspondence theorem.
• Free modules; rank; universal property; free modules over integral domains; the torsion submodule.
• Modules over a principal ideal domain; The classification of finitely generated modules over a principal ideal domain.
• The classification of finitely generated abelian groups.
• The Jordan normal form of matrices over the complex numbers.
• Matrix groups; general and special linear, orthogonal, symplectic and unitary groups; possibly a survey of crystallographic groups in dimensions 2 and 3.
• Representation theory for finite groups over the complex numbers; Schur’s Lemma, Maschke’s Theorem, character theory and examples of character tables in small examples.
• Burnside’s p-q Theorem
