Partition a group into cosets and Lagrange’s theorem falls out. Then build new groups from old ones via direct products, classify all finitely generated abelian groups, and see group theory at work in the plane isometries.

Sequence

1. Cosets and the Theorem of Lagrange — coset partitions and the divisibility law.
2. Direct Products and Finitely Generated Abelian Groups — building groups and the Structure Theorem.
3. Plane Isometries — translations, rotations, reflections, glide reflections.