Large orbits of nilpotent subgroups of linear groups

TL;DR

The paper studies orbit structures for nilpotent subgroups $H$ of finite solvable groups $G$ acting on finite faithful completely reducible $G$-modules $V$, establishing a quantitative bound on orbit sizes: for the smallest prime $p$ dividing $|H|$, there exists $v\in V$ with $|\mathbf{C}_H(v)| \le (|H|/p)^{1/p}$, equivalently $|v^H|^2 \ge 2|H|$ when $2\mid|H|$. It generalizes Isaacs's results from $p$-groups to nilpotent subgroups within solvable linear groups by leveraging the Hartley–Turull lemma and a minimal counterexample framework, and develops a detailed structural analysis via the Fitting subgroup and extra-special $p$-groups to bound orbit sizes. The approach systematically reduces to irreducible and primitive actions, combining theoretical bounds with computational verifications for exceptional cases. The findings yield concrete orbit-size bounds in the linear-group setting and pave the way for related results and applications in permutation and representation theory, including connections to Keller–Yang and related generalizations.

Abstract

Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $N$ be a nilpotent subgroup of $G$, then there exits $v \in V$ such that $|\bC_N(v)| \leq (|N|/p)^{1/p}$, where $p$ is the smallest prime divisor of $|N|$.

Theorems & Definitions (29)

• Definition 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.6