Monster group

Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the happy family, and the remaining six exceptions pariahs.

Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in Scientific American.

The character table of the monster, a 194-by-194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne.

In his 1982 paper, he referred to the monster as the Friendly Giant, but this name has not been generally adopted.

For example, the simple groups A100 and SL20(2) are far larger but easy to calculate with as they have "small" permutation or linear representations.

The documentation states that multiplication of group elements takes less than 40 milliseconds on a typical modern PC, which is five orders of magnitude faster than estimated by Robert A. Wilson in 2013.

[12][13][14][15] The mmgroup software package has been used to find two new maximal subgroups of the monster group.

[16] Previously, Robert A. Wilson had found explicitly (with the aid of a computer) two invertible 196,882 by 196,882 matrices (with elements in the field of order 2) which together generate the monster group by matrix multiplication; this is one dimension lower than the 196,883-dimensional representation in characteristic 0.

Performing calculations with these matrices was possible but is too expensive in terms of time and storage space to be useful, as each such matrix occupies over four and a half gigabytes.

[18] Wilson with collaborators found a method of performing calculations with the monster that was considerably faster, although now superseded by Seysen's abovementioned work.

This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together.

specifically between the nodes of the diagram and certain conjugacy classes in the monster, known as McKay's E8 observation.

These are the sporadic groups associated with centralizers of elements of type 1A, 2A, and 3A in the monster, and the order of the extension corresponds to the symmetries of the diagram.

See ADE classification: trinities for further connections (of McKay correspondence type), including (for the monster) with the rather small simple group PSL(2,11) and with the 120 tritangent planes of a canonic sextic curve of genus 4 known as Bring's curve.

The monster group is one of two principal constituents in the monstrous moonshine conjecture by Conway and Norton,[29] which relates discrete and non-discrete mathematics and was finally proved by Richard Borcherds in 1992.

"[31] Simon P. Norton, an expert on the properties of the monster group, is quoted as saying, "I can explain what Monstrous Moonshine is in one sentence, it is the voice of God.