Navarro vertices and lifts in solvable groups

Abstract

Let Q be a p-subgroup of a finite p-solvable group G, where p is a prime, and suppose that δ is a linear character of Q with the property that $\delta (x)=\delta (y)$ whenever $x,y\in Q$ are conjugate in G. In this situation, we show that restriction to p-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of G with Navarro vertex $(Q,\delta )$ onto the set of irreducible Brauer characters of G with Green vertex Q. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.

Communicated by Mandi Schaeffer Fry

Keywords:

• Brauer characters
• lifts
• linear characters
• Navarro vertices
• π-partial characters

2020 Mathematics Subject Classification:

• Primary: 20C15
• Secondary: 20C20

Acknowledgments

The authors would like to thank the referee for many valuable suggestions and corrections which have improved the exposition of the paper.

Additional information

Funding

This work was supported by the NSF of China (No. 12171289) and the NSF of Shanxi Province (Nos. 20210302123429, 20210302124077).