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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]], let $p$ be a prime, and let $P$ be a [[Sylow $p$-subgroup|0053CQE9Y2]] of $G$ with [[normaliser|0YGN3VAA0D]] $N_G(P)$.\nFor a finite group $H$, let $\\operatorname{Irr}_{p'}(H)$ be the set of irreducible complex [[characters|RMWMYBY8TC]] of $H$ whose degree is not divisible by $p$.\nThen\n$$\\abs{\\operatorname{Irr}_{p'}(G)} = \\abs{\\operatorname{Irr}_{p'}(N_G(P))}.$$\n\n## Remarks\n\nJohn McKay stated the conjecture in the early 1970s.\nIsaacs, Malle and Navarro reduced it in 2007 to a stronger condition on finite simple groups, the inductive McKay condition.\nMalle and Späth completed the case $p = 2$ in 2016.\nCabanes and Späth completed the proof for all primes in 2024, so the statement is now a theorem, though it keeps its name.\nExample: let $G = S_3$ and $p = 3$, with $P = A_3$ normal.\nThen $N_G(P) = G$, so both sides count the same set.\nFor $p = 2$ 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.\n\n## Explanations\n\n### L1 human\n\nThe conjecture compares a finite group with a much smaller subgroup, the normaliser of a Sylow subgroup.\nIt says that both have the same number of irreducible characters whose degree is not a multiple of $p$.\nSo a global count is determined by local data near the prime $p$.\n\n### L2 human\n\nIt is the prototype of the local-global conjectures of modular representation theory, with Alperin's weight conjecture and Dade's conjecture.\nFor $p$-solvable groups it was proved earlier, by Okuyama and Wajima.\nThe proof for all groups goes through the [[classification of finite simple groups|06J6P03MG7]] and the character theory of groups of Lie type, using [[Deligne–Lusztig theory|ZH21D5B62C]].\n",
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