2019/02/14 by Schulte, Egon, Soberón, Pablo, Williams, Gordon Ian
#52B15 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.1902.05439
We study finite groups that occur as combinatorial automorphism groups or geometric symmetry groups of convex polytopes. When Γ is a subgroup of the combinatorial automorphism group of a convex d-polytope, d≥ 3, then there exists a convex d-polytope related to the original polytope with combinatorial automorphism group exactly Γ. When Γ is a subgroup of the geometric symmetry group of a convex d-polytope, d≥ 3, then there exists a convex d-polytope related to the original polytope with both geometric symmetry group and combinatorial automorphism group exactly Γ. These symmetry-breaking results then are applied to show that for every abelian group Γ of even order and every involution σ of Γ, there is a centrally symmetric convex polytope with geometric symmetry group Γ such that σ corresponds to the central symmetry.