Directed power graph of a group
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Definition
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.