TITLE: What is a "natural group action"?
QUESTION [5 upvotes]: Eg. The symmetric group on S acts on S in a natural way, for all sets S.
Thanks in advance!

REPLY [2 votes]: For comparison's sake, we can try to build an "unnatural" group action.  Given a group $G$ and a set $S$, we know that the identity $e \in G$ must act as the identity map and that the action of two elements is the same as the action of their composition.  
I propose as our "unnatural action" we take $S_n$ as our group and $S$ to be a set of $n$ labeled elements.  Then, instead of $(1,2) \in S_n$ transposing the $1^{st}$ and $2^{nd}$ elements, it transposes the $5^{th}$ and $9^{th}$ elements.  There is certainly a way to make this a well-defined group action, but it doesn't feel natural at all.