The uniqueness of vertex pairs in $\pi$-separable groups

TL;DR

This paper investigates the uniqueness of vertex pairs in finite π-separable groups, introducing twisted vertices to establish conditions under which linear twisted vertices are unique, especially when 2 is not in π.

Contribution

It introduces the concept of twisted vertices for cases where 2 is not in π and proves their uniqueness under specific conditions, answering a previously posed question.

Findings

Linear twisted vertices are unique when χ is an N-lift or has a linear Navarro vertex.

The paper extends the understanding of vertex pair uniqueness beyond the case where 2 is in π.

It provides new tools for analyzing characters in π-separable groups.

Abstract

Let $G$ be a finite $\pi$-separable group, where $\pi$ is a set of primes, and let $\chi$ be an irreducible complex character that is a $\pi$-lift of some $\pi$-partial character of $G$.It was proved by Cossey and Lewis that all of the vertex pairs for $\chi$ 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 twisted vertices in the case where $2\notin\pi$, and establish the uniqueness for linear twisted vertices under the conditions that either $\chi$ is an $\mathcal N$-lift for a $\pi$-chain $\mathcal N$ of $G$ or it has a linear Navarro vertex, thus answering a question proposed by them.