Lifts of Brauer characters in characteristic two

Abstract

A conjecture raised by Cossey in 2007 asserts that if G is a finite p-solvable group and φ is an irreducible p-Brauer character of G with vertex Q, then the number of lifts of φ is at most |Q:Q′|. This conjecture is now known to be true in several situations for p odd, but there has been little progress for p even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift might be neither linear nor conjugate. In this paper we show that if χ is a lift of an irreducible 2-Brauer character in a solvable group, then χ has a linear Navarro vertex if and only if all the vertex pairs of χ are linear, and in that case all of the twisted vertices of χ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for p=2 “one vertex at a time”.