WOLFRAM|DEMONSTRATIONS PROJECT

Sporadic Groups

A group is a set of elements

that is closed under an associative binary operation such that

contains an identity element

and each element

has an inverse

-1

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

-1

for

g∈G

. (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

{e}

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.