Finite groups with few character values that are not character degrees

TL;DR

The paper investigates how the sets of character values ${\mathrm {cv}}(G)$ and ${\mathrm {cdc}}(G)$ constrain the structure of finite groups, extending BCZ95's work on small numbers of values. It establishes that if every non-linear irreducible character has at most four distinct values, then the group is solvable, and it provides a detailed classification for cases where the complement of character degrees, ${\mathrm {cdc}}(G)$, has size 2 or 3, including precise structures for solvable groups and notable results for non-solvable groups through rational extendible characters and factor group analysis. The paper also develops a framework of preliminary lemmas about roots of unity and value sets, then explores explicit examples and poses several open problems. Overall, it demonstrates that tight limits on character-values sets force strong solvability restrictions and illuminate nuanced non-solvable configurations, connecting to Frobenius, extraspecial, and rational-group phenomena.

Abstract

Let $ G $ be a finite group and $ χ\in \mathrm{Irr}(G) $. Define $ \mathrm{cv}(G)=\{χ(g)\mid χ\in \mathrm{Irr}(G), g\in G \} $, $ \mathrm{cv}(χ)=\{χ(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}(χ)|=3 $ for every non-linear $ χ\in \mathrm{Irr}(G) $ and their results show that $ G $ is solvable. They also considered groups in which $ |\mathrm{cv}(χ)|=4 $ for some non-linear $ χ\in \mathrm{Irr}(G) $. Continuing with their work, in this article, we prove that if $ |\mathrm{cv}(χ)|\leqslant 4 $ for every non-linear $ χ\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}(χ)|\leqslant 4 $ for all $ χ\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 $.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7