Derived length when a product of characters has two nonprincipal irreducible constituents

Abstract

We study the situation where a solvable group [Formula: see text] has a faithful irreducible character [Formula: see text] such that [Formula: see text] has exactly two distinct nonprincipal irreducible constituents. We prove that [Formula: see text] has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.