Every coprime linear group admits a base of size two

TL;DR

This paper proves that any coprime linear group acting on a finite vector space has a base of size at most two, confirming a conjecture and strengthening previous results in the area.

Contribution

It establishes the minimal base size for coprime linear groups as two, generalizing and improving prior bounds and answering a longstanding question.

Findings

Proved that coprime linear groups have base size at most two

Confirmed the sharpness of the bound

Connected the result to the k(GV) theorem and large orbit methods

Abstract

Let G be a linear group acting on the finite vector space V and assume that (|G|,|V|)=1. In this paper we prove that G has a base size at most two and this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative. Via large orbits this is related to the k(GV) theorem.