On the power graphs of finite groups and Hamilton cycle

The power graph P(G) of a finite group G is a graph whose vertex set is the group G and distinct elements x, y ∈ G are adjacent if one is a power of the other, that is, x and y are adjacent if x ∈ ⟨y⟩ or y ∈ ⟨x⟩. In this paper, we study existence of the Hamilton cycle in the power graph of some finite nilpotent groups G with a cyclic subgroup as direct factor when G is written as direct product Sylow p-subgroups. For this purpose we use of cartesian product a spanning tree and a cycle. Finally, we determined values of n such that P(Un) is Hamiltonian, where Un is a group consist of all positive integers less than n and relatively prime to n under multiplication modulo n.