Polytopes with Preassigned Automorphism Groups

TL;DR

This paper demonstrates that any finite group can be realized as the automorphism group of a finite abstract polytope, which can also be represented as a convex polytope, expanding the understanding of symmetry groups in polytope theory.

Contribution

It proves that every finite group can be realized as the automorphism group of a convex polytope or a face-to-face tessellation of a sphere by convex polytopes.

Findings

Every finite group is the automorphism group of some finite abstract polytope.

Such abstract polytopes can be realized as convex polytopes.

Abstract polytopes can tessellate spheres with convex polytopes.

Abstract

We prove that every finite group is the automorphism group of a finite abstract polytope isomorphic to a face-to-face tessellation of a sphere by topological copies of convex polytopes. We also show that this abstract polytope may be realized as a convex polytope.