Simple groups, fixed point ratios and applications

TL;DR

This survey explores fixed point ratios in simple groups of Lie type, emphasizing key results, applications, and open problems in permutation group theory.

Contribution

It provides a comprehensive overview of fixed point ratios in simple Lie type groups, summarizing recent advances and highlighting their significance.

Findings

Fixed point ratios are crucial in understanding permutation actions of simple groups.

The survey summarizes main results on fixed point ratios for groups of Lie type.

Applications include group recognition and probabilistic group theory.

Abstract

The study of fixed point ratios is a classical topic in permutation group theory, with a long history stretching back to the origins of the subject in the 19th century. Fixed point ratios arise naturally in many different contexts, finding a wide range of applications. In this survey article we focus on fixed point ratios for simple groups of Lie type, highlighting some of the main results, applications and related problems.