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 ${\displaystyle p}$ to that of the normalizer of a Sylow ${\displaystyle p}$-subgroup. It is named after Canadian mathematician John McKay.

Statement

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

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

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

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

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

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.