Abstract: Let G be a finite group and χ ∈ Irr(G). Define cv(G)={χ(g) | χ ∈ Irr(G),g∈ G}, cv(χ)={χ(g) | g ∈ G} and denote by dl(G) the derived length of G. In the 1990s Berkovich, Chillag and Zhmud described groups G in which |cv(χ)| =3for every non-linear χ ∈ Irr(G) and their results show that G is solvable. They also considered groups in which |cv(χ)| =4for some non-linear χ ∈ Irr(G). Continuing with their work, in this article, we prove that if |cv(χ)| ⩽ 4 for every non-linear χ ∈ 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)| ⩽ 4 or dl(G) ⩽ 3. It is interesting to note that these groups are such that |cv(χ)| ⩽ 4 for all χ ∈ 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.

Cite: Madanha S.Y. , Mbaale X. , Mudziiri Shumba T.M.
Finite groups with few character values that are not character degrees
Journal of Pure and Applied Algebra. 2025. V.229. N7. 107969 :1-15. DOI: 10.1016/j.jpaa.2025.107969 Scopus OpenAlex

Dates:

Submitted:	Jul 25, 2024
Accepted:	Mar 13, 2025
Published online:	Apr 11, 2025
Published print:	Jul 17, 2025

Identifiers:

Scopus:	2-s2.0-105002813339
OpenAlex:	W4409349056

Citing: Пока нет цитирований

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