TITLE: Can a group act on the empty set?
QUESTION [13 upvotes]: There isn't much more to add to this question. Can we define an action between some group and the null set?
I would have thought that there being no elements to act on it trivially satisfies the requirements for something to be an action but I'm not sure.

REPLY [14 votes]: 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.