The uniqueness of vertex pairs in π-separable groups

Abstract

Let G be a finite π-separable group, where π is a set of primes, and let χ be an irreducible complex character that is a π-lift of some π-partial character of G. It was proved by Cossey and Lewis that all of the vertex pairs for χ are linear and conjugate in G if $2\in \pi$, but the result can fail for $2\notin \pi$. In this paper we introduce the notion of the linear twisted vertices in the case where $2\notin \pi$, and then establish the uniqueness for such vertices under the conditions that either χ is an $\mathcal{N}$-lift for a π-chain $\mathcal{N}$ of G or it has a linear Navarro vertex, thus answering a question proposed by them.

Keywords:

• Lifts
• twisted vertices
• vertex pairs
• π-factored characters
• π-partial characters

2020 Mathematics Subject Classification:

• Primary: 20C15
• Secondary: 20C20

Acknowledgments

The authors would like to thank the referee for helpful comments.

Additional information

Funding

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