# Difference between revisions of "McKay conjecture"

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This article is about a conjecture in the following area in/related to group theory: representation theory. View all conjectures and open problems

## Statement

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

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

$|\operatorname{Irr}_{p'}(G)| = |\operatorname{Irr}_{p'}(N_G(P))|$

## Particular cases

The McKay conjecture holds trivially for finite nilpotent groups, because each Sylow subgroup is normal and hence $G = N_G(P)$. More generally, it holds whenever the $p$-Sylow subgroup is normal.

## Progress towards the conjecture

### For p-solvable groups

It has been proved that if $G$ is a $p$-solvable group, the McKay conjecture holds for the group $G$ and prime $p$. 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:

### 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.