Prime Divisibility Among Degrees of Solvable Groups

Abstract

Let G be a finite, nonabelian, solvable group. Following work by D. Benjamin, we conjecture that some prime must divide at least a third of the irreducible character degrees of G. Benjamin was able to show the conjecture is true if all primes divide at most 3 degrees. We extend this result by showing if primes divide at most 4 degrees, then G has at most 12 degrees. We also present an example showing our result is best possible.

Key Words:

• Character degree
• Solvable group

2000 Mathematics Subject Classification:

• 20C15

Acknowledgments

The author thanks Dr. Stephen Gagola and Dr. Donald White for their help in editing this document. Also, the author cannot express enough appreciation to his advisor, Dr. Mark Lewis, for indispensible guidance on this work, which is a portion of the author's doctoral dissertation. A special thanks is extended to Dr. I. M. Isaacs for help in streamlining the proof of the Main Theorem as well as several helpful insights.

Notes

^#Communicated by A. Turull.