${\rm B}_\pi$-characters and quotients

TL;DR

This paper investigates the properties of Isaacs ${ m B}_ ext{pi}$-characters in finite $ ext{pi}$-separable groups, establishing a relationship between characters of the group and its quotients.

Contribution

It proves that the set of ${ m B}_ ext{pi}$-characters of a quotient group equals the intersection of the group's ${ m B}_ ext{pi}$-characters with the irreducible characters of the quotient.

Findings

${ m B}_ ext{pi}(G/N) = { m Irr}(G/N) igcap { m B}_ ext{pi}(G)$ for normal subgroup $N$

Provides a characterization of ${ m B}_ ext{pi}$-characters in quotient groups

Enhances understanding of character theory in $ ext{pi}$-separable groups

Abstract

Let $\pi$ be a set of primes, and let $G$ be a finite $\pi$-separable group. We consider the Isaacs ${\rm B}_\pi$-characters. We show that if $N$ is a normal subgroup of $G$, then ${\rm B}_\pi (G/N) = {\rm Irr} (G/N) \cap {\rm B}_\pi (G)$.