Lifts and vertex pairs in solvable groups

TL;DR

This paper investigates the relationship between lifts of Brauer characters and vertex pairs in odd p-solvable groups, establishing conjugacy class results and conditions for vertex pairs.

Contribution

It introduces new results linking lifts of Brauer characters to vertex pairs, including conjugacy class formation and criteria for vertex pair realization.

Findings

Vertex pairs of a lift form a single conjugacy class

Provides a sufficient condition for a pair to be a vertex pair of a lift

Analyzes the behavior of lifts with respect to normal subgroups

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.