A note on derived length and character degrees

Authors

Keywords:

Solvable Groups, Derived Length, Real and Monolithic Characters, the Taketa Inequality and Isaacs-Seitz Conjecture

Abstract

Isaacs and Seitz have conjectured that the derived length of a finite solvable group G is bounded by the cardinality of the set of all irreducible character degrees of G, namely that the inequality $\textrm{dl}(G)\leq|\textrm{cd}(G)|$ holds for $G$, where $\textrm{dl}(G)$ is the derived length of $G$ and $\textrm{cd}(G)$ is the set of all irreducible character degrees of $G$. In this paper, we first prove that the inequality holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having same kernels are distinct. Also, we show that the conjecture is true in the case when $|\textrm{Irr}_{1, m}(G)|\leq 3$, where $\textrm{Irr}_{1, m}(G)$ is the set of all nonlinear monolithic characters of $G$. Additionally, we give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup $G'$ is supersolvable.

Published

2021-03-10

Issue

Vol. 103 No. 2 (2021)

Section

Articles
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