For example, consider a cube. You can rotate it in several different ways to produce an indistinguishable copy of the original; such an operation is called a symmetry. Doing two successive rotations is equivalent to doing some single rotation. In all, a cube has 24 rotational symmetry elements (not counting an equal number that involve reflections).

A tetrahedron has 12 rotational symmetry operations, which constitute a subgroup of the symmetries of the cube.

A normal subgroup of a group is defined by the property that for . (For some intuition about normal subgroups, see [2].) For example, is a normal subgroup of .

A group is simple if it has no normal subgroups apart from the trivial ones and itself. A simple group is like a prime number in arithmetic. Groups can be analyzed by repeatedly factoring out the largest normal subgroups.

There are 18 infinite families of finite simple groups plus 26 exceptional sporadic groups. This Demonstration provides binary generator matrices and for 23 of the 26 sporadic groups (the remaining three are too large). From these, a binary number can select a multiplication sequence of these generators.

An enormously complex proof of the theorem classifying all finite simple groups was tentatively completed by Daniel Gorenstein in 1983.