Monster group
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In mathematics, the Monster group M or IM (aka Fischer-Griess Monster, or The Friendly Giant) is a group of order
- 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
- = 808017424794512875886459904961710757005754368000000000
- ≈ 8 · 1053.
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; 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.
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[edit] Existence and uniqueness
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 196884-dimensional commutative nonassociative algebra. John Conway subsequently simplified this construction.
Griess's and Conway's constructions show that the Monster exists. John G. Thompson showed that uniqueness would follow from the existence of a 196883-dimensional faithful representation. A proof of the existence of such a representation was announced in 1982 by Simon P. Norton, but the details have not yet been published. The first published proof of uniqueness of the monster was completed by Griess, Meierfrankenfeld, and Segev in 1990.
The character table of the Monster was calculated in 1979, before the Monster was proven either to exist or be unique. The calculation is based on the assumption that the minimal degree of a faithful complex representation is 196883.
[edit] Moonshine
The Monster group prominently features in the Monstrous Moonshine conjecture which relates discrete and non-discrete mathematics and was proven by Richard Borcherds in 1992.
In this setting, the Monster is visible as the automorphism group of the Monster module, a vertex operator algebra, an infinite dimensional algebra containing the Griess algebra, and acts on the Monster Lie algebra, a generalized Kac-Moody algebra.
[edit] A computer construction
Robert A. 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 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).
[edit] See also
- Planetary (comics)#References describes a fictional use of the monster group.
[edit] References
- R. L. Griess, Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102
- Griess, Robert L., Jr.; Meierfrankenfeld, Ulrich; Segev, Yoav A uniqueness proof for the Monster. Ann. of Math. (2) 130 (1989), no. 3, 567-602.
- P. E. Holmes and R. A. Wilson, A computer construction of the Monster using 2-local subgroups, J. London Math. Soc. 67 (2003), 346--364.
- 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.
- Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England 1985.
- S. P. Norton, The uniqueness of the Fischer-Griess Monster, Finite groups---coming of age (Montreal, Que., 1982), 271--285, Contemp. Math., 45, Amer. Math. Soc., Providence, RI, 1985.
- J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308--339.
