Inducing $\pi$-partial characters with a given vertex

TL;DR

This paper investigates the induction of $pi$-partial characters in solvable groups, establishing bounds on the number of Brauer characters with a given vertex under specific group order conditions.

Contribution

It provides a new bound on the number of Brauer characters inducing a given character with a specified vertex in solvable groups, under conditions on the group order and prime.

Findings

Bound on the number of Brauer characters with a given vertex

Applicable when group order is odd or prime is 2

Generalizes previous results in character theory

Abstract

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup of a subgroup $V$. Suppose $\phi \in \ibr G$. If either $|G|$ is odd or $p = 2$, we prove that the number of Brauer characters of $H$ inducing $\phi$ with vertex $Q$ is at most $|\norm GQ: \norm VQ|$.