# Characters of $\pi'$-degree

## TL;DR

This paper proves that a finite group with only the trivial irreducible character of degree not divisible by at most two primes must be trivial, extending understanding of character degree restrictions.

## Contribution

It establishes a new characterization of trivial groups based on the absence of nontrivial irreducible characters with degrees coprime to a small set of primes.

## Key findings

- If $ ext{Irr}_{ ext{π'}}(G)$ contains only the principal character, then $G$ is trivial for sets of at most two primes.
- The result generalizes previous character degree restrictions to groups with limited prime sets.
- Provides a criterion for triviality of finite groups based on character degree properties.

## Abstract

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