Products of Irreducible Characters Having Complex-Valued Constituents

Abstract

First, we prove that when a finite solvable group $G$ has a faithful irreducible character $\chi$ such that $\chi\overline{\chi}$ has two irreducible constituents, both must be real-valued. Then, we study the situation where $\chi\overline{\chi}$ has exactly three distinct nonprincipal irreducible constituents, two of which are complex conjugates. In this case, we prove that $G$ has derived length bounded above by $6$.

Keywords

• product of characters
• derived length