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 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 $p$:

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

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 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:

• All the sporadic simple groups: This was proved by Wilson in his paper: The McKay conjecture is true for 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

• 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 2^{More 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 groups^{More info}
• The McKay conjecture is true for the sporadic simple groups by R. A. Wilson, Journal of Algebra, 207 (1998), Page 294-305: In this paper, Wilson settled the McKay conjecture for sporadic simple groups^{More info}
• Paper:IsaacsNavarro01^{More info}
• New refinements of the McKay conjecture for finite groups by I. Martin Isaacs and Gabriel Navarro, Annals of Mathematics, Volume 156, Page 333 - 344(Year 2002): ^{ArXiV copyMore info}