Character tables and Sylow normalizers

Abstract

Let $ \mathcal{Q} \in \mathrm{Syl}_q (G) $ , where G is a p-solvable group. We show that $ \mathbf{N}_{G}(\mathcal{Q}) $ is a p′-group if and only if each irreducible character of G of q′-degree is Brauer irreducible at the prime p. This result is generalized to $ \pi $ -separable groups, and one consequence, which can also be proved directly, is that the character table of a finite solvable group determines the set of prime divisors of the normalizer of a Sylow q-subgroup.