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
- 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 (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. These six exceptions are known as pariahs, and the others as the happy family.
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[edit] Existence and uniqueness
The Monster was predicted by Bernd Fischer and Robert Griess in 1973, as a simple group containing, as subquotients, the Fischer groups and some other sporadic simple groups. This group could not be precisely defined at first, but over the decade it was found possible to estimate the order of M and deduce other properties. Griess first constructed M 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 its uniqueness (as a simple group of the given order) 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 the uniqueness of M was completed by Griess, Meierfrankenfeld, and Segev in 1990.
The character table of the Monster, a 194-by-194 array, 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, product of the 3 largest prime divisors of |M|.
[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 group 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 (with elements in the field of order 2) which together 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] Subgroup structure
The Monster has at least 43 conjugacy classes of maximal subgroups. Non-abelian simple groups of some 60 isomorphism types are found as subgroups or as quotients of subgroups. The largest alternating group represented is A12. The Monster contains many but not all of the 26 sporadic groups as subgroups. This diagram, based on one in the book Symmetry and the Monster, shows how they fit together. The lines signify inclusion, as a subquotient, of the lower group by the upper one. The circled symbols denote groups not involved in larger sporadic groups. For the sake of clarity redundant inclusions are not shown.
[edit] In the arts
General features of the Monster, as well as some specific ones, were used to construct the musical composition Monsterology. The work was written for orchestra and computer-realized sound by Lawrence Fritts in 2004 and can be heard here: http://www.lawrencefritts.com/Monsterology.html.
[edit] References
- J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979), no. 3, 308--339.
- R. L. Griess, Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102
- 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.
- Griess, Robert L., Jr.; Meierfrankenfeld, Ulrich; Segev, Yoav A uniqueness proof for the Monster. Ann. of Math. (2) 130 (1989), no. 3, 567-602.
- 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.
- 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.
- M. Ronan, Symmetry and the Monster, Oxford University Press, 2006
