The subgroup graph of a group

Abstract

Given any subgroup H of a group G, let ΓH(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 ∈ H. Furthermore, if xy ∈ H and yx ∈ H for some x,y ∈ 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 ΓH(G) is investigated. All possible components are provided in the cases when |H| is either two or three, and the graph ΓH(G) is completely classified in the case when H is a normal subgroup of G and G/H is a finite abelian group. [Figure not available: see fulltext.].