Large orbits of Hall subgroups of solvable linear groups

TL;DR

The paper addresses bounding the $\pi$-part of the index $|G:O_{\pi'\pi}(G)|_{\pi}$ in finite solvable groups with Hall $\pi$-subgroups, relating it to the largest character degree $b(H)$ of the subgroup by proving $|G:O_{\pi'\pi}(G)|_{\pi} \leq b(H)^2$. It generalizes previous results by removing the requirement that $H$ be nilpotent and develops an orbit theorem for solvable linear groups, using module theory, reductions to irreducible/primitive cases, and modern tools from Holt–Dey and GAP to manage exceptional configurations. The main contribution is a universal bound tying orbit structure to a representation-theoretic invariant, with potential implications for orbit-size problems in linear group actions and MONA-type arguments. The work combines group-theoretic and representation-theoretic techniques to deepen understanding of Hall subgroups in solvable contexts.

Abstract

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $π$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{π' π}(G)|_π \leq b(H)^2$.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof