A converse for a theorem of Gallagher

TL;DR

This paper proves the converse of Gallagher's theorem for irreducible characters of finite groups, extends it to Isaacs' π-partial characters, and discusses related partial converses for the Brauer version.

Contribution

It establishes the converse of Gallagher's theorem, extends the result to π-partial characters, and explores partial converses for related theorems.

Findings

The converse of Gallagher's theorem holds for irreducible characters.

A partial converse of the Brauer version is proven.

An analog of Gallagher's theorem is valid for Isaacs' π-partial characters.

Abstract

Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\chi \in {\rm Irr} (G)$ satisfies $\chi_N$ is irreducible, then $\chi \beta$ is irreducible and distinct for all $\beta \in {\rm Irr} (G/N)$. Furthermore, if $\theta = \chi_N$, then these are all of the irreducible constituents of $\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\pi$-partial characters and that a partial converse of that theorem is true.