McKay conjecture

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 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 $Ir{r}_{p^{\prime }}(G)$ denotes the number of irreducible characters of $G$ whose order is not divisible by $p$:

$Ir{r}_{p^{\prime }}(G)=Ir{r}_{p^{\prime }}{N}_G(P)$

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

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:

• All the sporadic simple groups
• All the symmetric groups

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

• 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 groups^{More info}