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 $2\inπ$, but the result can fail for $2\notinπ$. In this paper we introduce the notion of the twisted vertices in the case where $2\notinπ$, and establish the uniqueness for linear twisted 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.