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 ${\displaystyle p}$ to that of the normalizer of a Sylow ${\displaystyle p}$-subgroup. It is named after Canadian mathematician John McKay.

Statement

Suppose ${\displaystyle p}$ is a prime number, ${\displaystyle G}$ is a finite group, and ${\displaystyle P\leq G}$ is a Sylow ${\displaystyle p}$-subgroup. Define

${\displaystyle {\textrm {Irr}}_{p'}(G):=\{\chi \in {\textrm {Irr}}(G):p\nmid \chi (1)\}}$

where ${\displaystyle {\textrm {Irr}}(G)}$ denotes the set of complex irreducible characters of the group ${\displaystyle G}$. The McKay conjecture claims the equality

${\displaystyle |{\textrm {Irr}}_{p'}(G)|=|{\textrm {Irr}}_{p'}(N_{G}(P))|}$

where ${\displaystyle N_{G}(P)}$ is the normalizer of ${\displaystyle P}$ in ${\displaystyle G}$.

References