Counting lifts of irreducible Brauer characters

TL;DR

This paper generalizes a previous result by providing an explicit description of the lifts of irreducible Brauer characters with a given vertex pair in $p$-solvable groups, under weaker conditions.

Contribution

It extends the 2011 result by Cossey, Lewis, and Navarro to cases where the vertex subgroup is not necessarily normal, offering a more general understanding.

Findings

Explicit description of lifts with a given vertex pair

Generalization of previous bounds on the number of lifts

Weaker conditions on the vertex subgroup Q

Abstract

Let $p$ be an odd prime, and suppose that $G$ is a $p$-solvable group and $\varphi\in {\rm IBr}(G)$ has vertex $Q$. In 2011, Cossey, Lewis and Navarro proved that the number of lifts of $\varphi$ is at most $|Q:Q'|$ whenever $Q$ is normal in $G$. In this paper, we present an explicit description of the set of lifts of $\varphi$ with a given vertex pair $(Q,\delta)$ under a weaker condition on $Q$, and thus generalize their result.