McKay conjecture

In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number $p$ to that of the normalizer of a Sylow $p$-subgroup. It is named after Canadian mathematician John McKay.

Statement

Suppose $p$ is a prime number, $G$ is a finite group, and $P\leq G$ is a Sylow $p$-subgroup. Define

${\textrm {Irr}}_{p'}(G):=\{\chi \in {\textrm {Irr}}(G):p\nmid \chi (1)\}$

where ${\textrm {Irr}}(G)$ denotes the set of complex irreducible characters of the group $G$. The McKay conjecture claims the equality

$|{\textrm {Irr}}_{p'}(G)|=|{\textrm {Irr}}_{p'}(N_{G}(P))|$

where $N_{G}(P)$ is the normalizer of $P$ in $G$.

References

• Isaacs, I.M. (1994). Character Theory of Finite Groups. Dover. ISBN 0-486-68014-2. (Corrected reprint of the 1976 original, published by Academic Press.)
• Evseev, Anton (2013). "The McKay Conjecture and Brauer's Induction Theorem". Proceedings of the London Mathematical Society. 106: 1248–1290. arXiv:1009.1413. doi:10.1112/plms/pds058.