A group is a set of elements that is closed under an associative binary operation such that contains an identity element and each element in has an inverse .[more]
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 .) 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.[less]
The generators of a group are a set of elements such that every element of the group can be expressed as a product of a finite number of generators. For example, this Demonstration has 10×10 matrices and over as generators for the 7920 elements of the Mathieu group . (The binary operation is matrix multiplication, and is the Galois field with two elements, 0 and 1.) For element 7920, the decimal sequence for the product is 85283493; the equivalent binary number is 101000101010101001010100101. Substituting for 1 and for 0 gives the product .
The 23 sporadic groups in this Demonstration are:
These three sporadic groups were judged to be too large for this Demonstration:
Matrix representations are from .
 R. A. Wilson, R. A. Parker, and J. N. Bray, "ATLAS of Finite Group Representations." (Apr 13, 2016) for.mat.bham.ac.uk/atlas/html/contents.html.
 J. Baez, "What's a Normal Subgroup?" (Apr 13, 2016) math.ucr.edu/home/baez/normal.html.