The number of Hamiltonian cycles in groups of Symmetry

Groups are algebraic structures in Abstract Algebra comprised of a set of elements with a binary operation that satisfies closure, associativity, identity and invertibility. Cayley graphs serve as a visualization tool for groups, as they are capable of illustrating certain structures and properties of groups geometrically. In particular, each element in a group is assigned to a vertex in Cayley graphs. By the group action of left-multiplication, distinct elements in the generating sets can act on each element to create varied directed edges (Meier[1]). By contrast, the presence of a Hamiltonian cycles within a graph demonstrates its level of connectivity. In this research paper, utilizing directed Cayley graphs, we present a series of conjectures and theorems regarding the number and existence of Hamiltonian cycles within Dihedral groups, Symmetric groups of Platonic solids and Symmetric groups. By exploring the relationship between Abstract groups and Hamiltonian graphs, this work contributes to the broader field of research pertaining to Groups of Symmetry and Geometric Group Theory.