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 $G$.

History

In McKay's original papers on the subject , the statement was given for the prime $p=2$ but examples of computations of $|{\textrm {Irr}}_{p'}(G)|$ for odd primes are also mentioned. The first appearance of the conjecture for arbitrary primes is in a paper by Jon L. Alperin giving also a version in block theory, now called the Alperin-McKay conjecture

Proof

In 2007, Marty Isaacs, Gunter Malle and Gabriel Navarro showed that the McKay conjecture reduces to the checking of a so-called inductive McKay condition for each finite simple group . This opens the door to a proof of the conjecture by using the classification of finite simple groups.

The paper of Isaacs-Malle-Navarro was also an inspiration for similar reductions for Alperin weight conjecture, its block version, the Alperin-McKay conjecture and Dade's conjecture.

The McKay conjecture for the prime 2 was proven by Gunter Malle and Britta Späth in 2016 .

An important step in proving the inductive McKay condition for all simple groups is to determine the action of the group of automorphisms ${\textrm {Aut}}(G)$ on the set ${\textrm {Irr}}(G)$ for each finite quasisimple group $G$. The solution has been announced by Späth in .

The McKay conjecture for all primes and all finite groups was announced by Marc Cabanes and Britta Späth in October 2023.