Characters of 𝜋’-degree

Abstract

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