Monster group

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In the mathematical field of group theory, the Monster group M or F1 (also known as the Fischer-Griess Monster, or the Friendly Giant) is a group of finite order

It is a simple group, meaning it does not have any normal subgroups except for the subgroup consisting only of the identity element, and M itself.

The finite simple groups have been completely classified (the classification of finite simple groups). The list of finite simple groups consists of 18 countably infinite families, plus 26 sporadic groups that do not follow such a systematic pattern. The Monster group is the largest of these sporadic groups and contains all but six of the other sporadic groups as subquotients. Robert Griess has called these six exceptions pariahs, and refers to the others as the happy family.

Contents

Existence and uniqueness

The Monster was predicted by Bernd Fischer (unpublished) and Robert Griess (1976) in about 1973 as a simple group containing a double cover of Fischer's baby monster group as a centralizer of an involution.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. Griess (1982) constructed M as the automorphism group of the Griess algebra, a 196884-dimensional commutative nonassociative algebra. John Conway and Jacques Tits subsequently simplified this construction.

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