McKay conjecture

Remarks

John McKay stated the conjecture in the early 1970s. Isaacs, Malle and Navarro reduced it in 2007 to a stronger condition on finite simple groups, the inductive McKay condition. Malle and Späth completed the case $p = 2$ in 2016. Cabanes and Späth completed the proof for all primes in 2024, so the statement is now a theorem, though it keeps its name. Example: let $G = S_3$ and $p = 3$, with $P = A_3$ normal. Then $N_G(P) = G$, so both sides count the same set. For $p = 2$ and $G = S_3$, take $P = \set{1, (1\,2)}$; then $N_G(P) = P$ has $2$ characters of odd degree, and $S_3$ has the characters of degrees $1, 1, 2$, so $2$ of odd degree.

Going deeper

For a student

It is the prototype of the local-global conjectures of modular representation theory, with Alperin's weight conjecture and Dade's conjecture. For $p$-solvable groups it was proved earlier, by Okuyama and Wajima. The proof for all groups goes through the classification of finite simple groups and the character theory of groups of Lie type, using Deligne–Lusztig theory.