nLab
Monster group

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Contents

Idea

The Monster group M is a finite group that is the largest of the sporadic finite simple groups. It has order

2 46⋅3 20⋅5 9⋅7 6⋅11 2⋅13 3⋅17⋅19⋅23⋅29⋅31⋅41⋅47⋅59⋅71 =808017424794512875886459904961710757005754368000000000\begin{aligned} & 2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71 \\ & = 808017424794512875886459904961710757005754368000000000 \end{aligned}

and contains all but six of the other 25 sporadic finite simple groups as subquotients.

See also Moonshine.

History

The Monster group was predicted to exist by Bernd Fischer and Robert Griess in 1973, as a simple group containing the Fischer groups? and some other sporadic simple groups as subquotients. Subsequent work by Fischer, Conway, Norton and Thompson estimated the order of M and discovered other properties and subgroups, assuming that it existed. In a famous paper

Griess proved the existence of the largest simple sporadic group. The author constructs “by hand” a non-associative but commutative algebra of dimension 196883, and showed that the automorphism group of this algebra is the conjectured friendly giant/monster simple group. The name “Friendly Giant” for the Monster did not take on.

After Griess found this algebra Igor Frenkel, James Lepowsky and Meurman and/or Borcherds showed that the Griess algebra is just the degree 2 part of the infinite dimensional Moonshine vertex algebra.

David Roberts: wasn’t there some discussion on the cafe about the sporadic groups potentially being part of more systematic families of weaker structures? (monoids or something) If there are any references in that discussion they could perhaps be pulled in here and at classification of finite simple groups and/or sporadic finite simple group.

The Monster

Revised on July 19, 2012 18:54:23 by Urs Schreiber (131.220.203.120)
Edit | Back in time (8 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: group, group theory, Ab, abelian group, group algebra, 2-group, cat-n-group, finite group, semidirect product group, topological group, k-tuply groupal n-groupoid, group object, torsion subgroup, subquotient, Lie group, compact Lie group, quasigroup, Hochschild-Serre spectral sequence, group cohomology, n-group, strict 2-group, groupoid object in an (infinity,1)-category, Nielsen-Schreier theorem, free group, braid group, normal subgroup, normalizer, orbit, general linear group, special orthogonal group, orthogonal group, group extension, loop group, unitary group, local Lie group, subgroup, infinity-group, coset, Poisson Lie group, Monster group, sporadic finite simple group, classification of finite simple groups, simple group, Robert Griess, Bernd Fischer, Lie 2-group, special unitary group, E8, Lie group cohomology, symplectic group, mapping class group, Heisenberg group, exceptional Lie group, Lorentz group, projective unitary group, circle n-group, symmetry, group theory - contents, Gruppenpest, group of units, locally compact topological group, divisible group, p-divisible group, iterated loop space object, Picard group, contents of contents, Moufang loop, G2, 3-group, spin^c, group of order 2, code loop, maximal compact subgroup, anti de Sitter group, Picard 3-group, sylleptic 3-group, symmetric 2-group, abelian infinity-group, induced character, infinity-group of units, conjugacy class, infinite-dimensional Lie group, almost connected topological group, Coxeter group, E6, E7, compact topological group, identity component, k-monoidal infinity-group, abelian 3-group, spin representation, Kac-Moody group, order of a group, metaplectic group, Baer sum, maximal torus, group of Lie type, p-torsion, Mathieu group, cyclic group, geometric group theory, free abelian group, infinity-group extension, commutator subgroup, abelianization, group commutator, bilinear map, tensor product of abelian groups, Feit-Thompson theorem, augmentation ideal, quotient group, anabelian group, Cauchy group theorem, index of a subgroup, Galois cohomology, finite abelian group, braided 2-group, braided infinity-group, braided 3-group, symmetric 3-group, smooth 2-group, Picard 2-group, Picard infinity-group, Weyl group, binary Golay code, abelian 4-group, abelian 7-group, cosmic Galois group, motivic Galois group