nLab Monster group



Exceptional structures

Group Theory



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

2 463 205 97 611 213 3171923293141475971 =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 (the ‘pariah groups’) of the other 25 sporadic finite simple groups as subquotients, called the Happy Family.

See also Moonshine.


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 MM 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.

There is a school of thought, going back to at least Israel Gelfand, that sporadic groups are really members of some other infinite families of algebraic objects, but due to numerical coincidences or the like, just happen to be groups (see this nCafe post). One version of this, in the case of the Monster (and perhaps for other sporadic groups via Moonshine phenomena) is that what we know as the Monster is just a shadow of a 2-group, as the Monster can be constructed as an automorphism group of a conformal field theory, a structure rich enough to have a automorphism 2-group(oid) (see this nCafe discussion).


Via Coxeter groups

The Monster admits a reasonably succinct description in terms of Coxeter groups. Let [n][n] denote the linear graph with vertices 0,1,,n0, 1, \ldots, n with an edge between adjacent numbers i,i+1i, i+1 and no others. If 11 is the terminal (1-element) graph, there is a map 0:1[n]0: 1 \to [n], mapping the vertex of 11 to the vertex 00. Regarding this as an object in the undercategory 1Graph1 \downarrow Graph, let Y 443Y_{443} be the coproduct of the three objects 0:1[4]0: 1 \to [4], 0:1[4]0: 1 \to [4], 0:1[3]0: 1 \to [3] in 1Graph1 \downarrow Graph. This (pointed) graph has 12 elements and is shaped like a YY, with arms of length 4, 4, 3 emanating from a central vertex of valence 33.

Regard Y 443Y_{443} as a Coxeter diagram. The associated Coxeter group C 443C_{443} is given by a group presentation with 12 generators (represented by the vertices) of order 22 (so 12 relators of the form x 2=1x^2 = 1), with a relation (xy) 3=1(x y)^3 = 1 if x,yx, y are adjacent vertices (so 11 relators, one for each edge), and xy=yxx y = y x if x,yx, y are non-adjacent (55 more relators). This Coxeter group (12 generators, 78 relators) is infinite, but by modding out by another strange ‘spider’ relator

(ab 1c 1ab 2c 2ab 3c 3) 10=1(a b_1 c_1 a b_2 c_2 a b_3 c_3)^{10} = 1

the resulting quotient NN turns out to be a finite group. Here aa is the central vertex of valence 33, b 1,c 1b_1, c_1 are on an arm of length 44 with b 1b_1 adjacent to aa and c 1ac_1 \neq a adjacent to b 1b_1; similarly for b 2,c 2b_2, c_2 on the other arm of length 44, and for b 3,c 3b_3, c_3 on the arm of length 33. See here if this is not clear.

It turns out that NN has a center CC of order 22, and the Monster MM is the quotient, i.e. the indicated term in the exact sequence

1CNM1.1 \to C \to N \to M \to 1.

This implicitly describes the Monster in terms of 12 generators and 80 relators.

Such “YY-group” presentations (Coxeter group based on a similar YY-diagram, modulo a spider relation) are linked to a number of finite simple group constructions, the most famous of which is perhaps Y 555Y_{555} which is a presentation of the “Bimonster” (the wreath product of the Monster with /2\mathbb{Z}/2). See Ivanov for a general description of these. The presentation of the Monster given above was established in Ivanov2.

Via automorphisms of a super vertex operator algebra

There is a super vertex operator algebra, the Monster vertex operator algebra, whose group of of automorphisms of a VOA is the monster group.

(Frenkel-Lepowski-Meurman 89, Griess-Lam 11)


Possible relation to bosonic M-theory:

Discussion of aspects of the Monster group via (Platonic) 2-groups:

The Monster

Last revised on July 1, 2024 at 21:14:35. See the history of this page for a list of all contributions to it.