Prescribing Symmetries and Automorphisms for Polytopes

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