Prescribing Symmetries and Automorphisms for Polytopes

We study the groups for which it is possible to find a convex polytope with that group as automorphism group with additional geometric conditions on the action of the group or its subgroups. In particular, we prove that for every abelian group G of even order and an involution s of G, there is a centrally symmetric convex polytope whose automorphism group is G and such that s corresponds to the central symmetry.