A dual version of Huppert's rho-sigma conjecture for character codegrees

TL;DR

This paper classifies finite groups where any two distinct character codegrees are coprime and proposes a conjecture relating the number of character codegrees divisible by primes to the prime divisors of the group order, proving it for solvable groups.

Contribution

It introduces a classification of groups with coprime character codegrees and proves a related conjecture for solvable groups.

Findings

Classification of groups with coprime character codegrees

Conjecture relating codegree divisibility to prime divisors

Proof of the conjecture for solvable groups

Abstract

We classify the finite groups with the property that any two different character codegrees are coprime. In general, we conjecture that if $k$ is a positive integer such that for any prime $p$ the number of character codegrees of a finite group $G$ that are divisible by $p$ is at most $k$, then the number of prime divisors of $|G|$ is bounded in terms of $k$. We prove this conjecture for solvable groups.