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

Navarro, Gabriel; Tiep, Pham; Turull, Alexandre

doi:10.1090/S1088-4165-07-00263-4

$p$-rational characters and self-normalizing Sylow $p$-subgroups
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by Gabriel Navarro, Pham Huu Tiep and Alexandre Turull

Represent. Theory 11 (2007), 84-94

DOI: https://doi.org/10.1090/S1088-4165-07-00263-4

Published electronically: April 19, 2007

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Abstract:

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

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