Large orbits of nilpotent subgroups of linear groups

TL;DR

This paper investigates the action of nilpotent subgroups within finite solvable groups on modules, establishing bounds on the size of centralizers of elements, which advances understanding of group-module interactions.

Contribution

It provides a new bound on the size of centralizers of elements in nilpotent subgroups acting on modules, extending previous results in group theory.

Findings

Existence of a vector with bounded centralizer size in the module

Bound depends on the smallest prime divisor of the subgroup order

Enhances understanding of nilpotent subgroup actions in solvable groups

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 $|N|$.