# Can a group act on the empty set?

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.

• Though it's kind of empty to have a group action on an empty set, isn't it? =) Mar 10, 2019 at 11:30
• In particular, the symmetric group $S_0$, which has order $1$, acts naturally on the empty set. There is unique bijection between the empty set and itself. Mar 10, 2019 at 11:57
• @user21820 the interest of a mathematical formalism is to avoid such philosophical considerations. In the same spirit, there were mathematicians fighting against the existence of infinite sets in the late XIX...
– YCor
Mar 10, 2019 at 13:41
• @YCor: Erm... I was just joking in my first comment, but I disagree with your comment, because anyone who claims they use ZFC as their foundational system necessarily has made some very weird philosophical assumptions whether or not they know it. Mar 10, 2019 at 14:10
• @YCor: But that's only if you think "truth within set theory" is meaningful. To refrain from prolonging this thread with our off-topic discussion, do you want to come to the logic chat-room? Mar 10, 2019 at 14:16

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