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

TL;DR

This paper classifies certain non-solvable groups based on the prime divisors of their non-linear character degrees, identifying specific groups and prime restrictions.

Contribution

It characterizes non-solvable groups with non-linear character degrees sharing the same number of prime divisors, extending previous solvable group results.

Findings

Identifies specific non-solvable groups fitting the criteria.

Shows only primes 2, 3, 5, 7 divide their irreducible character degrees.

Confirms Huppert's ρ-σ conjecture for these groups.

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 $\rho$-$\sigma$ conjecture holds for them.