View source for McKay conjecture

{{conjecture|representation theory}}

==Statement==

Let <math>G</math> be a finite group and <math>p</math> a [[prime number]] dividing the order of <math>G</math>. Let <math>P</math> be a <math>p</math>-[[Sylow subgroup]] of <math>G</math>. Then the number of irreducible complex characters of <math>G</math> whose degree is not divisible by <math>p</math>, equals the number of irreducible complex characters of <math>N_G(P)</math>, whose degree is not divisible by <math>p</math>.

In other words, if <math>\operatorname{Irr}_{p'}(G)</math> denotes the set of irreducible characters of <math>G</math> whose degree is not divisible by <math>p</math>:

<math>|\operatorname{Irr}_{p'}(G)| = |\operatorname{Irr}_{p'}(N_G(P))|</math>

==Generalizations and related conjectures==

* [[Alperin-McKay conjecture]] is a version of the conjecture that gives a more precise statement about a bijection for each block.
* [[Relative McKay conjecture]]

==Particular cases==

The McKay conjecture holds trivially for [[finite nilpotent group]]s, because each Sylow subgroup is normal and hence <math>G = N_G(P)</math>. More generally, it holds whenever the <math>p</math>-Sylow subgroup is normal.

==Progress towards the conjecture==

===For p-solvable groups===

It has been proved that if <math>G</math> is a <math>p</math>-solvable group, the McKay conjecture holds for the group <math>G</math> and prime <math>p</math>. This was shown by Okuyama and Wajima in 1979, in their paper ''Irreducible characters of p-solvable groups'' (see References for more).

Some stronger versions of the McKay conjecture, including the Alerpin-McKay conjecture and the relative McKay conjecture, have been verified for these groups.

===For some infinite families of almost simple groups===

The McKay conjecture has not been checked for all finite simple groups, but it has been checked for some, notably:

* All the [[sporadic simple group]]s: This was proved by Wilson in his paper: ''The McKay conjecture is true for sporadic simple groups''
* All the [[symmetric group]]s

===Reductions===

In addition to work done to check the McKay conjecture for some particular groups, a lot of work has been done to reduce the general McKay conjecture to some specific cases, using the method of finding minimal counterexamples.

==References==
* {{paperlink|IntroMcKayconjecture}}
* {{paperlink|OkWaj79}}
* {{paperlink|Wilson98}}
* {{paperlink|IsaacsNavarro01}}
* {{paperlink|IsaacsNavarro02}}