a group is one of the simplest but most powerful ideas in mathematics. a group is just a set of elements with an operation that follows four rules. 1. closure combine any two elements → the result is still inside the set. 2. identity there exists a special element that changes nothing. a • e = a 3. inverse every element has another element that cancels it. a • a⁻¹ = e 4. associativity the order of grouping does not matter. (a • b) • c = a • (b • c) that’s it. no geometry. no numbers required. yet this simple structure describes symmetry everywhere: • rotations of a square • permutations of objects • transformations in physics • cryptography systems...