On the number of conjugacy classes of zeros of characters and the length of solvable groups

Abstract

We apply an orbit theorem to a few questions about character degrees. We investigate the relation of the number of conjugacy classes where characters vanish and the length of the solvable groups. As another application, we give a bound for the size of defect groups of blocks of solvable groups.

Communicated by J. Zhang

Keywords:

• Character degrees
• conjugacy classes

2000 Mathematics Subject Classification:

• 20C15; 20C20

Acknowledgment

The author is grateful to the referee for valuable suggestions and comments.

Additional information

Funding

The project was supported by NSFC (Nos: 11671063) and a grant from the Simons Foundation (#499532).