Sylow Normalizers and Brauer Character Degrees

Abstract

Suppose that G is a finite group. In this note, we show that a local condition about Sylow normalizers is equivalent to a global condition on the degrees of certain irreducible Brauer characters of G. Theorem A. Let G be a finite ”p; q•-solvable group, and let Q ∈ SylqG‘ and P ∈ SylpG‘. Then every irreducible p-Brauer character of G of q′degree has p′-degree if and only if NGQ‘ is contained in some G-conjugate of NGP‘. Theorem A needs a solvability hypothesis. If p = 7, then the irreducible p-Brauer characters of the group G = PSL2; 27‘ have degrees ”1; 13; 26; 28•. If we set q = 2, then each q′-degree is also a p′-degree.