6533b7d7fe1ef96bd1268365

RESEARCH PRODUCT

𝑝-rational characters and self-normalizing Sylow 𝑝-subgroups

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Let G G be a finite group, p p a prime, and P P a Sylow p p -subgroup of G G . Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of p ′ p’ -degree of G G and the irreducible characters of p ′ p’ -degree of N G ( P ) \mathbf {N}_G(P) , which preserves field of values of correspondent characters (over the p p -adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If p > 2 p>2 , then G G has no non-trivial p ′ p’ -degree p p -rational irreducible characters if and only if N G ( P ) = P \mathbf {N}_G(P)=P .