General Prerequisites: Knowledge of the first and second-year algebra courses is helpful but not mandatory; in particular Prelims M1: Groups and Group Actions, A0: Linear Algebra, and ASO: Group Theory. Likewise, the course B3.5 Topology and Groups would bring more familiarity and a different viewpoint of the notions treated in this course.

Course Overview: The course introduces some natural families of groups, with an emphasis on those that generalize abelian groups, various questions that one can ask about them, and various methods used to answer these questions; these involve among other things questions of finite presentability, linearity, torsion and growth.

Course Synopsis: Free groups; ping-pong lemma. Finitely generated and finitely presented groups. Residual finiteness and linearity.

Nilpotency, lower and upper central series. Polycyclic groups, length, Hirsch length, Noetherian induction. Solvable groups, derived series, law.

Structure of linear solvable and nilpotent groups.

Solvable versus polycyclic: examples, characterization of polycyclic groups as solvable Z-linear groups.

Solvable versus nilpotent: the Milnor -Wolf theorem characterizing nilpotent groups as solvable groups with sub-exponential (and in fact polynomial) growth.