# Difference between revisions of "McKay conjecture"

## 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 order is not divisible by $p$, equals the number of irreducible complex characters of $N_G(P)$, whose order is not divisible by $p$.

In other words, if $Irr_{p'}(G)$ denotes the number of irreducible characters of $G$ whose order is not divisible by $p$:

$Irr_{p'}(G) = Irr_{p'}N_G(P)$

## 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.

## References

• A new invariant for simple groups by John McKay, Notices Amer. Math. Soc. 18 (1971), 397: In this paper, McKay introduced a special case of the McKay conjecture, for the situation of a finite simple group, where the prime equals 2More info
• Irreducible characters of p-solvable groups by Tetsuro Okuyama and Masayuki Wajima, Proc. Japan. Academy, 55, Ser. A, 1979: This paper settles the McKay conjecture for p-solvable groupsMore info