# Groups with few $p'$-character degrees

## TL;DR

This paper proves a variation of Thompson's Theorem showing that certain finite groups with restricted character table properties are trivial, using the classification of finite simple groups.

## Contribution

It introduces a new variation of Thompson's Theorem relating character table properties to the triviality of specific group substructures.

## Key findings

- Groups with two distinct non-divisible character table entries are trivial.
- Uses classification of finite simple groups to establish the result.
- Provides a new criterion for group triviality based on character table analysis.

## Abstract

We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group $G$ contains only two distinct values not divisible by a given prime number $p>3$, then $O^{pp'pp'}(G)=1$. This is done by using the classification of finite simple groups.