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.

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 $G$ .

Isaacs, I.M. (1994). Character Theory of Finite Groups (Corrected reprint of the 1976 original, published by Academic Press. ed.). Dover. ISBN 0-486-68014-2.
Evseev, Anton (2011). "The McKay Conjecture and Brauer's Induction Theorem". arXiv:1009.1413 [math.RT].

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