Mathematical Research Seminar - Archive

In this talk we adopt the notion of skeletal polytope introduce by Grunbaum in the second half of the 20th Century. A polygon is then a connected 2-valent graph embedded in Euclidean space (no assumption on convexity or planarity), and in general an n-polytope is a collection of (n-1)-polytopes satisfying certain axioms. Such a structure is called chiral if it admits all possible symmetry by abstract rotations, but none by abstract reflections.

Chiral polyhedra (polytopes of rank 3) were found and classified by Schulte only in 2005. Chiral 4-polytopes were claimed not to exist in 2004. This last claim was mistaken. In this talk we present the three chiral 4-polytopes in space.

A group is called 2-genetic if each normal subgroup of the group can be generated by two elements. Let G be a non-abelian 2-genetic group of order p^n for an odd prime p and a positive integer n. In this paper, we investigate connected Cayley digraphs Cay(G, S), and determine their full automorphism groups when Aut(G, S) = {α\in Aut(G) | S^α = S} is a p′-group. With the result, we give the first known half-arc-transitive non-normal Cayley graphs of order an odd prime-power.