Large orbits of nilpotent subgroups of solvable linear groups
Volume 180 / 2026
Colloquium Mathematicum 180 (2026), 213-222
MSC: Primary 20C20; Secondary 20C15, 20D10
DOI: 10.4064/cm9653-1-2026
Published online: 7 May 2026
Abstract
Suppose that $G$ is a finite solvable group and $V$ is a finite, faithful and completely reducible $G$-module. Let $H$ be a nilpotent subgroup of $G$. Then there exists $v \in V$ such that $|\mathbf C_H(v)| \leq (|H|/p)^{1/p}$, where $\mathbf C_H(v)$ is the centralizer of $v$ in $H$ and $p$ is the smallest prime divisor of $|H|$.
