Non-solvable groups whose non-linear character degrees have the same number of different prime divisors

Abstract

By a result of Noritzsch, a finite solvable group whose non-linear character degrees have the same set of prime divisors is meta-abelian. In this note we investigate finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors, and show that up to an abelian direct factor, such groups are exactly $L_2(4), L_2(8), A_7, S_7$, the central product of a cyclic $3$-group with $3.A_7$, or the semi-direct product of $A_7$ by a cyclic $2$-group $\langle a\rangle$ such that $a$ non-trivially acts on $A_7$ by conjugation. As consequence, we show that only the primes $2,3,5,7$ may occur as prime divisors of their irreducible character degrees, and that Huppert's $ρ$-$σ$ conjecture holds for them.