Directed power graph of a group
Let be a group. The directed power graph of is a directed graph whose vertices are the elements of and where there is an edge from a vertex to a vertex if is a power of . Note that this graph contains loops at every point, though we can modify the definition to avoid loops.
Note that there is an edge from to and an edge from to if and only if and are powers of each other.
The directed power graph of a group is a combinatorial datum about the group and the power graph, up to graph isomorphism, determines the group up to 1-isomorphism.