Group Theory and Symmetry

A group is a set with one operation and four rules, and that deliberate poverty is what lets a single theorem cover rotations, shuffles, clock arithmetic and Rubik's cube at once. This path starts from the symmetries of a square, proves Lagrange's theorem and gets Fermat's little theorem for free, shows why no crystal can have five-fold symmetry and what happened when one did, and ends with the group of order 60 that makes a quintic formula impossible.