On Hamiltonian Path and Circuits in Non-Abelian Finite Groups

The main objective of this paper is to determine the non-Abelian finite groups which contain only Abelian and Hamiltonian subgroups and to obtain some of their fundamental properties. Two exceptional groups of orders 16 and 24 were examined and are completely determined using GAP. These were achieved from the fact that if a group G contains at least one Hamiltonian subgroup and if all its subgroups are Abelian or Hamiltonian, then the group itself is Hamiltonian. We finally generate some Hamiltonian circuits in the two non-Abelian groups and then present a method of finding the number of circuits in any finite group.