Directed power graph of a group

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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.

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Facts