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