Directed power graph of a group

Definition
Let $$G$$ be a group. The directed power graph of $$G$$ is a directed graph whose vertices are the elements of $$G$$ and where there is an edge from a vertex $$g$$ to a vertex $$h$$ if $$h$$ is a power of $$g$$. 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 $$g$$ to $$h$$ and an edge from $$h$$ to $$g$$ if and only if $$g$$ and $$h$$ 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.

Related notions

 * Undirected power graph of a group

Facts

 * Finite groups are 1-isomorphic iff their directed power graphs are isomorphic
 * Undirected power graph determines directed power graph for finite group