Automorphisms of classical geometries in the sense of Klein

TL;DR

This paper computes the automorphism groups of classical geometries as defined by Klein and applies these results to construct an outer automorphism of the symmetric group S_6.

Contribution

It provides explicit calculations of automorphism groups for Klein's geometries and demonstrates an application to symmetric group automorphisms.

Findings

Automorphism groups of projective, affine, and Euclidean geometries are explicitly determined.

A simple construction of the outer automorphism of S_6 is presented.

The results deepen understanding of symmetries in classical geometries.

Abstract

In this note, we compute the group of automorphisms of Projective, Affine and Euclidean Geometries in the sense of Klein. As an application, we give a simple construction of the outer automorphism of S_6.