The finite simple groups

A simple group is one that can't be broken into smaller groups: its only normal subgroups are the trivial one and itself. Every finite group is assembled from simple ones, the way molecules are assembled from atoms, so these are the atoms of symmetry. The classification says the full list is on this page: two families you can describe in a line, sixteen families built from Lie theory, and 26 sporadic groups that fit no pattern. The proof runs to tens of thousands of pages spread over hundreds of papers.

On the map: group theory, Galois theory, the Sylow theorems, simple Lie algebras, the classification, the Monster and moonshine.

Each tile is an infinite family, with its smallest simple member and a trace of how fast the orders grow over its first eight members. Tap one for the list. The diagram on each Lie tile is its Dynkin diagram; the tile below is the same family twisted by a symmetry of that diagram.

The 26 sporadic groups

Groups that belong to no family. Twenty of them live inside the largest, the Monster, in three generations; the six pariahs do not. The bar under each is its order on a log scale, with the Monster at full width.