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{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.

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