A converse for a theorem of Gallagher

TL;DR

This work establishes a converse to Gallagher's theorem in the Brauer-character setting and derives its ordinary-character corollary, highlighting a necessity via a counterexample when $p$ does not divide $|G/N|$. It then generalizes to Isaacs' $\pi$-theory, proving a Gallagher-type bijection for $B_{\pi}$-characters under the conditions $2\in\pi$ or $|G|$ odd, and develops a broader $\pi$-version using nuclei from normal chains. The paper further develops a general, parity-free $\pi$-version of Gallagher's theorem, with a detailed construction of nuclei and a bijection framework for irreducible $\pi$-partial characters, while also presenting partial converses in the $\pi$-setting and corresponding lift-based corollaries. Overall, the results illuminate the limits and scope of Gallagher-type phenomena across ordinary, Brauer, and $\pi$-theoretic character theories, offering new tools (nuclei) and clarifying when full converses are possible.

Abstract

Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $χ\in {\rm Irr} (G)$ satisfies $χ_N$ is irreducible, then $χβ$ is irreducible and distinct for all $β\in {\rm Irr} (G/N)$. Furthermore, if $θ= χ_N$, then these are all of the irreducible constituents of $θ^G$. 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' $π$-partial characters and that a partial converse of that theorem is true.

Theorems & Definitions (31)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• proof : Proof of Theorem \ref{['main2']}
• proof : Proof of Theorem \ref{['main1']}
• Example 2.1
• Lemma 4.1
• proof