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Encyclopedia > Monster group

In mathematics, the Monster group M is a group of order Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ...

   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. In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...


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. In mathematics the term countable set is used to describe the size of a set, e. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ... The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the 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 196884-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. 1980 is a leap year starting on Tuesday. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. ... In mathematics, a vertex operator algebra (abbreviated: VOA) is a certain kind of algebra that plays a key part in conformal field theory and other fields of study in physics, and has also proven useful in purely mathematical contexts such as moonshine theory. ... In mathematics, a generalized Kac-Moody algebra is a Lie algebra that is similar to a Kac-Moody algebra, except that it is allowed to have imaginary simple roots. ...


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). In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0 (where n1R is defined as 1R + ... + 1R with n summands). ...


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 mathematics, monstrous moonshine is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the (then totally unexpected) connection between the monster group M and modular functions (particularly, the j function). ... Richard Ewen Borcherds (born November 29, 1959) is a mathematician specializing in group theory and Lie algebras. ...


References

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.


Atlas of monster representations


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Monster group - Wikipedia, the free encyclopedia (452 words)
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.
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.
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.
  More results at FactBites »

 
 

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