Finite groups with few character values that are not character degrees

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