Groups with few $p^{\prime }$-character degrees

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 $O^{p{p}^{\prime }p{p}^{\prime }}(G)=1$. This is done by using the classification of finite simple groups.