Characters of ${\pi }^{\prime }$-degree

Abstract

Let $G$ be a finite group and let $\pi$ be a set of primes. Write ${\mathrm{Irr}}_{{\pi }^{\prime }}(G)$ for the set of irreducible characters of degree not divisible by any prime in $\pi$. We show that if $\pi$ contains at most two prime numbers and the only element in ${\mathrm{Irr}}_{{\pi }^{\prime }}(G)$ is the principal character, then $G=1$.