# A Geometric Characterization of Rational Groups

## TL;DR

This paper provides a geometric criterion to identify finite rational groups by relating their structure to automorphisms of a finite geometry, offering a new perspective on their classification.

## Contribution

It introduces a geometric characterization of finite rational groups based on their action on a finite geometry and flag fixation properties.

## Key findings

- Finite rational groups can be characterized by their automorphism actions on finite geometries.
- A group is rational iff it acts on a geometry with conjugate elements fixing the same number of flags.
- Provides a new geometric perspective for understanding the structure of rational groups.

## Abstract

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\Gamma$ of type $I$ and action of $G$ on $\Gamma$ as a group of automorphisms such that if $g$ and $h$ are elements of $G$ fixing the same number of flags of type $J$ for all subsets $J$ of $I$, then $g$ and $h$ are conjugate in $G$.