Groups with some cube-free irreducible character co-degrees

Abstract

For a character χ of a finite group G, the number $\text{cod}(\chi )=\frac{[G:\text{ker}\chi ]}{\chi (1)}$ is called the co-degree of χ. Let $n\ge 2$ be an integer and $\text{Irr}(G|\text{Fit}(G))$ denote the set of irreducible characters whose kernels do not contain $\text{Fit}(G)$. In this paper, we show that if G is solvable and $p^{n+1}\nmid \text{cod}(\chi )$ for every prime divisor p of $\left|G\right|$ and every $\chi \in \text{Irr}(G|\text{Fit}(G))$, then the derived length of G is at most $2{\hspace{0.17em}\log \hspace{0.17em}}_2(n)+{\hspace{0.17em}\log \hspace{0.17em}}_2(n-1)+4$. Then, we classify the finite non-solvable groups with non-trivial Fitting subgroups such that the co-degrees of their irreducible characters whose kernels do not contain the Fitting subgroups are cube-free.

Keywords:

• Co-degree of a character
• Fitting subgroup
• the derived length of a solvable group

2020MATHEMATICS SUBJECT CLASSIFICATION:

• 20C15
• 20D25