Can a group act on the empty set?

group-theory group-actions

yes you can define the trivial action.

Note that the axioms for group action begins with "for all"

That is:

For all $x\in \emptyset$ we have that $e.x=x$.

For all $x\in\emptyset$ and all $g,h\in G$ we have $(gh)x=g.(h.x)$

Both statements hold trivially.

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