Bounding the Derived Length of a Solvable Group: An Improvement on a Result by Gluck

Abstract

Let cd(G) be the set of degrees of irreducible complex characters and dl(G) be the derived length of the finite group G. It is a result by Gluck that if G is solvable then dl(G) ≤2|cd(G)|. Using a result of Isaacs and Knutson, I will in this article show that this bound can be improved to dl(G) ≤2|cd(G)| −3 when |cd(G)| ≥3.

Key Words:

• Character degrees
• Derived length
• Solvable groups

2010 Mathematics Subject Classification:

• 20C15

ACKNOWLEDGMENT

I would like to thank Mark L. Lewis and J\ootrn B. Olsson for reading the preprint of this article and providing me with helpful comments.

The author is supported by QGM (Centre for Quantum Geometry of Moduli Spaces) funded by the Danish National Research Foundation.

Notes

Communicated by P. Tiep.