Finite groups with few character values that are not character degrees
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2025
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| author | Madanha, Sesuai Y. Mbaale, X. Shumba, Tendai M. Mudziiri |
| author_facet | Madanha, Sesuai Y. Mbaale, X. Shumba, Tendai M. Mudziiri |
| contents | 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 $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_24217 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite groups with few character values that are not character degrees Madanha, Sesuai Y. Mbaale, X. Shumba, Tendai M. Mudziiri Group Theory 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 $. |
| title | Finite groups with few character values that are not character degrees |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.24217 |