Monster group

In the mathematical field of group theory, the Monster group M or IM (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 are believed to have been completely classified (the classification of finite simple groups), although some doubts remain as to whether the completeness of this classification has been completely proven. Assuming the classification is correct, 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 subgroups or subquotients.

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 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 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.

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.

A computer construction

Robert A. Wilson has found explicitly (with the aid of a computer) two 196882 by 196882 matrices over the field with two 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).

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.

See also

• Planetary (comics)#References describes a fictional use of the monster group.

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.

External links

• MathWorld: Monster Group
• Atlas of Finite Group Representations: Monster group