The j-Function and the Monster

The first published notion that the j-function was in any way related to the Monster came in 1979, when Conway and Norton noted in [CN79] that each coefficient in the q- expansion of the j-function could be written as a (nontrivial) integral linear combination of the dimensions of irreducible representations of the Monster. For example, the first relevant coefficient, 196884, can be written 196883+1, where 196883 is the dimension of the smallest non-trivial representation of the Monster, and 1 is the dimension of the trivial representation.\ud This observation led to their conjecture, which they titled "monstrous moonshine." Here we give an introduction to "monstrous moonshine.