Monster group

In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, the friendly giant, or simply the Monster) is the largest sporadic simple group; it has order

The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. 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.

It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in Scientific American.

History

The monster was predicted by Bernd Fischer (unpublished, about 1973) and Robert Griess as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution. Within a few months, the order of M was found by Griess using the Thompson order formula, and Fischer, Conway, Norton and Thompson discovered other groups as subquotients, including many of the known sporadic groups, and two new ones: the Thompson group and the Harada–Norton group. The character table of the monster, a 194 × 194 array, was calculated in 1979 by Fischer and Donald Livingstone using computer programs written by Michael Thorne. It was not clear in the 1970s whether the monster actually existed.