Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675–686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).

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

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 .

Sylow subgroups and the number of conjugacy classes of p-elements