Lifts and vertex pairs in solvable groups

Abstract

Suppose $G$ is a $p$-solvable group, where $p$ is odd. We explore the connection between lifts of Brauer characters of $G$ and certain local objects in $G$, called vertex pairs. We show that if $\chi$ is a lift, then the vertex pairs of $\chi$ form a single conjugacy class. We use this to prove a sufficient condition for a given pair to be a vertex pair of a lift and to study the behavior of lifts with respect to normal subgroups.