So what the McKay conjecture is saying is this.

Suppose I'm interested in representing a Group as matrices over the complex numbers. There are usually many ways of doing this. Each one of them has a so-called character, which is like fingerprint of such a representation.

Along another line, it has been known that all groups contain large subgroup having an order which is a power of a prime--call it P. This group in turn has a normalizer in which P is normal--call it N(P).

The surprising thing is that the number of characters of G and of N(P)--which is is only a small part of G--is equal.

*technical note in both cases we exclude representation the degree of which is a multiple of p.