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.