McKay conjecture
# 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 $G$.

