Any Finite Group is the Group of Some Binary, Convex Polytope

TL;DR

This paper proves that every finite group can be represented as the automorphism group of a special convex polytope, specifically a binary, combinatorial polytope with a diameter at most 2, strengthening previous results.

Contribution

It provides a shorter proof and a stronger result showing all finite groups are automorphism groups of binary, combinatorial polytopes with specific geometric and combinatorial properties.

Findings

Constructs binary convex polytopes for any finite group

Shows automorphisms are induced by space isometries

Diameter of the polytope's skeleton is at most 2

Abstract

For any given finite group, Schulte and Williams (2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the convex polytope we build for the given finite group is binary, and even combinatorial in the sense of Naddef and Pulleyblank (1981); the diameter of its skeleton is at most 2; any combinatorial automorphism of the polytope is induced by some isometry of the space; any automorphism of the skeleton is a combinatorial automorphism.