# Groups satisfying the two-prime hypothesis with a composition factor isomorphic to ${\rm PSL}_2(q)$ for $q\geq 7$

## TL;DR

This paper investigates the structure of finite groups with a specific prime-related property in their character degrees, providing bounds on their irreducible character degrees when they include a certain simple group as a composition factor.

## Contribution

It establishes an upper bound on the number of irreducible character degrees for nonsolvable groups with a composition factor isomorphic to ${\rm PSL}_2(q)$ for $q \geq 7$, under the two-prime hypothesis.

## Key findings

- Provides an upper bound on irreducible character degrees.
- Characterizes groups with a ${\rm PSL}_2(q)$ composition factor.
- Extends understanding of prime-related properties in group character theory.

## Abstract

Let $G$ be a finite group, and write ${\rm cd}(G)$ for the degree set of the complex irreducible characters of $G$. The group $G$ is said to satisfy the {\it two-prime hypothesis} if, for any distinct degrees $a, b \in {\rm cd}(G)$, the total number of (not necessarily different) primes of the greatest common divisor ${\rm gcd}(a, b)$ is at most $2$. In this paper, we prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to ${\rm PSL}_2 (q)$ for $q \geq 7$.