McKay conjecture

In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number $$p$$ to that of the normalizer of a Sylow $p$-subgroup. It is named after Canadian mathematician John McKay.

Statement
Suppose $$p$$ is a prime number, $$G$$ is a finite group, and $$P \leq G$$ is a Sylow $$p$$-subgroup. Define
 * $$\textrm{Irr}_{p'}(G) := \{\chi \in \textrm{Irr}(G) : p \nmid \chi(1) \}$$

where $$\textrm{Irr}(G)$$ denotes the set of complex irreducible characters of the group $$G$$. The McKay conjecture claims the equality
 * $$|\textrm{Irr}_{p'}(G)| = |\textrm{Irr}_{p'}(N_G(P))|$$

where $$N_G(P)$$ is the normalizer of $$P$$ in $$G$$.