Groups with few 𝑝’-character degrees in the principal block

Abstract

Let p ≥ 5 p\ge 5 be a prime and let G G be a finite group. We prove that G G is p p -solvable of p p -length at most 2 2 if there are at most two distinct p ′ p’ -character degrees in the principal p p -block of G G . This generalizes a theorem of Isaacs–Smith as well as a recent result of three of the present authors.