McKay conjecture - Alchetron, The Free Social Encyclopedia

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 ≤ G is a Sylow p -subgroup. Define

Irr p ′ ( G ) := { χ ∈ Irr ( G ) : p ∤ χ ( 1 ) }

where Irr ( G ) denotes the set of complex irreducible characters of the group G . The McKay conjecture claims the equality

| Irr p ′ ( G ) | = | Irr p ′ ( N G ( P ) ) |

where N G ( P ) is the normalizer of P in G .

References

McKay conjecture Wikipedia

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