Degrees of irreducible characters and normal 𝑝-complements

Abstract

John Tate [1] proved that if P ∈ S y l p ( G ) P \in {\mathbf {S}}{\text {y}}{{\text {l}}_p}(G) , H H is a normal subgroup of a finite group G G and P ∩ H ≤ Φ ( P ) P \cap H \leq \Phi (P) ( Φ ( G ) \Phi (G) is the Frattini subgroup of G G ) then H H has a normal p p -complement. We prove in this note that Tate’s theorem has nice character-theoretic applications.