Can a group act on the empty set?

Question

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.

Answer 1

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.