Finite groups with few character values that are not character degrees
Abstract
Let G be a finite group and ∈ Irr(G) . Define cv(G)=\(g) ∈ Irr(G), g∈ G \ , cv()=\(g) g∈ G \ and denote dl(G) by the derived length of G . In the 1990s Berkovich, Chillag and Zhmud described groups G in which |cv()|=3 for every non-linear ∈ Irr(G) and their results show that G is solvable. They also considered groups in which |cv()|=4 for some non-linear ∈ Irr(G) . Continuing with their work, in this article, we prove that if |cv()|≤slant 4 for every non-linear ∈ Irr(G) , then G is solvable. We also considered groups G such that |cv(G) cd(G)|=2 . T. Sakurai classified these groups in the case when |cd(G)|=2 . We show that G is solvable and we classify groups G when |cd(G)|≤slant 4 or dl(G)≤slant 3 . It is interesting to note that these groups are such that |cv()|≤slant 4 for all ∈ Irr(G) . Lastly, we consider finite groups G with |cv(G) 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 dl(G)=2 .
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.