Ankur Moitra teaches this lecture from MIT's 18.200, Principles of Discrete Applied Mathematics, covering the foundations of group theory. He defines what a group is and works through examples including addition and multiplication modulo n, then introduces subgroups and cosets. The core of the session is a full statement and proof of Lagrange's theorem, relating the size of a subgroup to the size of its parent group. Moitra closes by applying Lagrange's theorem to prove Fermat's little theorem, showing how an abstract algebraic result yields a concrete number theory fact. The pace is that of a standard blackboard proof lecture, building definitions step by step before using them, suited to students who already have the prerequisites for a discrete math course and want a rigorous, example-driven treatment of basic group theory.