Products of Irreducible Characters Having Complex-Valued Constituents

• Main Authors: Lisa R. Hendrixson, Mark L. Lewis
• Format: Article
• Language: English
• Published: Aracne 2017-06-01
• Series: Advances in Group Theory and Applications
• Subjects: product of characters; derived length
• Online Access: http://www.advgrouptheory.com/journal/Volumes/4/L.R.%20Hendrixson,%20M.L.%20Lewis%20-%20Products%20of%20irreducible%20characters%20having%20complex-valued%20constituents.pdf

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 $6$.