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The finite simple groups have been completely classified; there are 18 countably infinite families of finite simple groups, plus 26 sporadic groups that do not follow any apparent pattern. The Monster group is the largest of these sporadic groups. See classification of finite simple groups.
The Monster was predicted by Bernd Fischer and Robert Griess in 1973, and first constructed by Griess in 1980 as the automorphism group of the Griess algebra, a 196883-dimensional commutative nonassociative algebra. The monster is also the automorphism group of the monster vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the monster Lie algebra, a generalized Kac-Moody algebra.
A computer construction
Robert Wilson has found explicitly (with the aid of a computer) two 196882 by 196882 matrices over the field with 2 elements that generate the Monster group. However, performing calculations with these matrices is prohibitively expensive in terms of time and storage space. Wilson with collaborators has found a method of performing calculations with the Monster that is considerably faster.
Let V be a 196882 dimensional vector space over the field with 2 elements. A large subgroup H (preferably a maximal subgroup) of the Monster is selected in which it is easy to perform calculations. The subgroup H chosen is 31+12.2.Suz.2, where Suz is the Suzuki simple group. Elements of the Monster are stored as words in the elements of H and an extra generator T. It is reasonably quick to calculate the action of one of these words on a vector in V. Using this action, it is possible to perform calculations (such as the order of an element of the Monster). Wilson has exhibited vectors u and v whose joint stabilizer is the trivial group. Thus (for example) one can calculate the order of an element g of the Monster by finding the smallest i > 0 such that giu = u and giv = v.
This and similar constructions (in different characteristics) have been used to prove some interesting properties of the Monster (for example, to find some of its non-local maximal subgroups).
S. A. Linton, R. A. Parker, P. G. Walsh and R. A. Wilson, Computer construction of the Monster, J. Group Theory 1 (1998), 307-337.
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