Navarro vertices and lifts in solvable groups

TL;DR

This paper establishes a canonical bijection between certain irreducible ordinary characters and Brauer characters in $p$-solvable groups, using Navarro vertices and linear characters, and explores their behavior under normal subgroups.

Contribution

It introduces a new correspondence linking Navarro vertices with Green vertices in $p$-solvable groups, enhancing understanding of character lifts and their properties.

Findings

Established a bijection between irreducible characters with Navarro vertices and Brauer characters with Green vertices.

Analyzed how lifts of Brauer characters behave with respect to normal subgroups.

Provided tools for studying character theory in $p$-solvable groups.

Abstract

Let $Q$ be a $p$-subgroup of a finite $p$-solvable group $G$, where $p$ is a prime, and suppose that $\delta$ is a linear character of $Q$ with the property that $\delta(x)=\delta(y)$ whenever $x,y\in Q$ are conjugate in $G$. In this situation, we show that restriction to $p$-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of $G$ with Navarro vertex $(Q,\delta)$ onto the set of irreducible Brauer characters of $G$ with Green vertex $Q$. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups.