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 [math]\displaystyle{ p }[/math] to that of the normalizer of a Sylow [math]\displaystyle{ p }[/math]-subgroup. It is named after Canadian mathematician John McKay.
Statement
Suppose [math]\displaystyle{ p }[/math] is a prime number, [math]\displaystyle{ G }[/math] is a finite group, and [math]\displaystyle{ P \leq G }[/math] is a Sylow [math]\displaystyle{ p }[/math]-subgroup. Define
- [math]\displaystyle{ \textrm{Irr}_{p'}(G) := \{\chi \in \textrm{Irr}(G) : p \nmid \chi(1) \} }[/math]
where [math]\displaystyle{ \textrm{Irr}(G) }[/math] denotes the set of complex irreducible characters of the group [math]\displaystyle{ G }[/math]. The McKay conjecture claims the equality
- [math]\displaystyle{ |\textrm{Irr}_{p'}(G)| = |\textrm{Irr}_{p'}(N_G(P))| }[/math]
where [math]\displaystyle{ N_G(P) }[/math] is the normalizer of [math]\displaystyle{ P }[/math] in [math]\displaystyle{ G }[/math].
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. doi:10.1112/plms/pds058.
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