# Prescribing Symmetries and Automorphisms for Polytopes

## TL;DR

This paper demonstrates how to prescribe specific symmetry and automorphism groups for convex polytopes, including constructing centrally symmetric polytopes with desired symmetry groups, advancing the understanding of polytope symmetries.

## Contribution

It establishes methods to realize any subgroup of automorphisms or symmetries as the exact symmetry group of a convex polytope, including centrally symmetric cases.

## Key findings

- Existence of convex polytopes with prescribed automorphism groups
- Construction of polytopes with exact geometric symmetry groups
- Realization of centrally symmetric polytopes with specific symmetry properties

## Abstract

We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When $\Gamma$ is a subgroup of the combinatorial automorphism group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with combinatorial automorphism group exactly $\Gamma$. When $\Gamma$ is a subgroup of the geometric symmetry group of a convex $d$-polytope, $d\geq 3$, then there exists a convex $d$-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly $\Gamma$. These symmetry-breaking results then are applied to show that for every abelian group $\Gamma$ of even order and every involution $\sigma$ of $\Gamma$, there is a centrally symmetric convex polytope with geometric symmetry group $\Gamma$ such that $\sigma$ corresponds to the central symmetry.