Spherical space forms revisited

TL;DR

This paper provides a simplified proof and slight improvement of Wolf's classification of finite groups acting freely and isometrically on spheres, linking them to Frobenius complements in finite group theory.

Contribution

It offers a more straightforward proof and refines the classification of groups acting on spheres, connecting geometric actions with algebraic structures.

Findings

Simplified proof of Wolf's classification

Refined classification removing redundancies

Identified groups as Frobenius complements

Abstract

We give a simplified proof of J. A. Wolf's classification of finite groups that can act freely and isometrically on a round sphere of some dimension. We slightly improve the classification by removing some non-obvious redundancy. The groups are the same as the Frobenius complements of finite group theory.