Classification of finite groups generated by reflections and rotations

TL;DR

This paper completes the classification of finite groups generated by reflections and rotations in Euclidean spaces, which are important in the study of orbifolds and quotient spaces.

Contribution

It extends existing partial classifications to a complete classification of these groups, enhancing understanding in geometric group theory.

Findings

Complete classification of these finite groups

Connection to orbifold theory and quotient spaces

Framework for analyzing symmetry groups in Euclidean spaces

Abstract

We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean space by a finite orthogonal group and hence in the theory of orbifolds.