Conway–Norton Moonshine conjecture

Let $V=\bigoplus_{n=0}^\fty V_n$ be a graded infinite-dimensional $\mathbb C$-vector space with $\dim V_n<\infty$ and a natural algebraic structure whose automorphism group is the Monster group. Each element $g$ of the Monster group acts linearly on every $V_n$. Conway–Norton Moonshine conjecture. For every $g$, the Laurent series

$$\sum_{n=0}^\infty ({\mathrm{Tr}}\;g|_{V_n})q^{n-1}$$

is a Hauptmodul corresponding to a genus $0$ subgroup of $SL(2,\mathbb R)$; for the identity element, it is the $j$-function without its constant term. This conjecture initiated the theory of Monstrous Moonshine; the stated Hauptmodul property is now a theorem, so the conjecture is solved.