Monster group

Remarks

The smallest nontrivial irreducible representation of $\mathbb{M}$ over $\C$ has dimension $196883$. Robert Griess constructed $\mathbb{M}$ as the automorphism group of a commutative non-associative algebra of dimension $196884$, the Griess algebra. Its existence had been predicted earlier by Bernd Fischer and by Griess. The $j$-invariant has the expansion $j(\tau) = q^{-1} + 744 + 196884 q + \cdots$ with $q = e^{2 \pi i \tau}$, and $196884 = 196883 + 1$. This coincidence led to the monstrous moonshine conjecture of Conway and Norton, proved by Richard Borcherds. Twenty of the $26$ sporadic groups are subquotients of $\mathbb{M}$, that is, quotients of subgroups of $\mathbb{M}$; the other six are the pariah groups.