The subgroup graph of a group

Given any subgroup H of a group G, let Γ H (G) be the directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if and only if x ≠ y and ${xy\in H}$ . Furthermore, if ${xy\in H}$ and ${yx\in H}$ for some ${x,y\in G}$ with x ≠ y, then x and y will be regarded as being connected by a single undirected edge. In this paper, the structure of the connected components of Γ H (G) is investigated. All possible components are provided in the cases when |H| is either two or three, and the graph Γ H (G) is completely classified in the case when H is a normal subgroup of G and G/H is a finite abelian group.