Finite groups with few character values that are not character degrees

Abstract

Let G be a finite group and $\chi \in \mathrm{Irr}\left(G\right)$. Define $\mathrm{cv}\left(G\right)=\left\{\chi \right(g\left)\right|\chi \in \mathrm{Irr}\left(G\right),g\in G\}$, $\mathrm{cv}\left(\chi \right)=\left\{\chi \right(g\left)\right|g\in G\}$ and denote by $\mathrm{dl}\left(G\right)$ the derived length of G. In the 1990s Berkovich, Chillag and Zhmud described groups G in which $\left|\mathrm{cv}\right(\chi \left)\right|=3$ for every non-linear $\chi \in \mathrm{Irr}\left(G\right)$ and their results show that G is solvable. They also considered groups in which $\left|\mathrm{cv}\right(\chi \left)\right|=4$ for some non-linear $\chi \in \mathrm{Irr}\left(G\right)$. Continuing with their work, in this article, we prove that if $\left|\mathrm{cv}\right(\chi \left)\right|⩽4$ for every non-linear $\chi \in \mathrm{Irr}\left(G\right)$, then G is solvable. We also considered groups G such that $\left|\mathrm{cv}\right(G)\setminus \mathrm{cd}(G\left)\right|=2$. T. Sakurai classified these groups in the case when $\left|\mathrm{cd}\right(G\left)\right|=2$. We show that G is solvable and we classify groups G when $\left|\mathrm{cd}\right(G\left)\right|⩽4$ or $\mathrm{dl}\left(G\right)⩽3$. It is interesting to note that these groups are such that $\left|\mathrm{cv}\right(\chi \left)\right|⩽4$ for all $\chi \in \mathrm{Irr}\left(G\right)$. Lastly, we consider finite groups G with $\left|\mathrm{cv}\right(G)\setminus \mathrm{cd}(G\left)\right|=3$. For nilpotent groups, we obtain a characterization which is also connected to the work of Berkovich, Chillag and Zhmud. For non-nilpotent groups, we obtain the structure of G when $\mathrm{dl}\left(G\right)=2$.