On the Orbits of Solvable Linear Groups

TL;DR

This paper investigates the structure of solvable linear groups acting on finite vector spaces, proving the existence of elements with trivial centralizer intersection, thus answering a question by Isaacs.

Contribution

It establishes the existence of elements with trivial centralizer intersection in solvable linear groups, extending previous results and addressing a specific open question.

Findings

Existence of x,y in V with C_G(x) ∩ C_G(y) = 1

Answers a question posed by Isaacs

Completes previous partial results by Dolphi, Seress, and Wolf

Abstract

Let $G$ be a solvable linear group acting on the finite vectorpace $V$ and assume that $(|G|,|V|)=1$. In this paper we find $x,y\in V$ such that $C_G(x)\cap C_G(y)=1$. In particular, this answers a question of I. M. Isaacs. We complete some results of S. Dolphi, A. Seress and T. R. Wolf.