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

Abstract

For any given finite group, Schulte and Williams (Discrete Comput Geom 54(2):444–458, 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 (J Combin Theory Ser B 31(3):297–312, 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.