group theory

Contents

Idea

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

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

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.

Revised on July 19, 2012 18:54:23 by Urs Schreiber (131.220.203.120)