Isaacs' Generalization of Taketa's theorem

TL;DR

The paper extends Taketa's solvability result for $M$-groups to Brauer and $\pi$-partial character settings by introducing pseudo $M_p$- and pseudo $M_\pi$-groups. It proves Isaacs-style generalizations: if every irreducible Brauer (or $\pi$-partial) character of a group $G$ is induced from a subgroup whose quotient lies in a chosen class $\mathcal{F}$ and certain $O$-conditions hold, then $G\in\mathcal{F}$. It further establishes solvability of pseudo $M_p$- and pseudo $M_\pi$-groups, and shows closure properties under normal Hall subgroups, while linking pseudo-monomiality to monomiality in $q$-power degrees. These results broaden the structural understanding of solvability and monomiality in modular and $\pi$-theoretic character theory, with connections to Dornhoff-type results and questions about the existence of nontrivial pseudo groups.

Abstract

We generalize the definition of pseudo monomial characters and $M$-groups to the Brauer character and Isaacs' $π$-partial character settings. We prove an analogs of Isaacs's generalization of Taketa's theorem in those settings. We consider other analogs of results regarding $M$-groups in those settings.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Lemma 2.1
• proof
• Theorem 2.2
• proof : Proof of Theorems \ref{['theorem1']} and \ref{['theorem2']}
• Theorem 3.1
• proof : Proof of Theorems \ref{['pseudo M_p']} and \ref{['pseudo M_pi']}