With the group axioms in hand, this part asks: what smaller groups live inside a given group, and how can we describe the simplest ones? The subgroup criterion, cyclic groups, and generating sets answer those questions.

Sequence

1. Subgroups — the $x{y}^{-1}$ criterion and intersections of subgroups.
2. Cyclic Groups — groups generated by a single element; order and generators.
3. Generators and Cayley Digraphs — finitely generated groups and graph visualisation.