The symmetric group $S_n$ for $n \in \mathbb N$ is a very well-studied object in mathematics. However, aside from Cayley's theorem I have never seen much discussion of the group of permutations over an infinite number of objects. Has there been any serious study or even definitions of groups like $S_{\aleph_0}$, the group of permutations of $\aleph_0$ elements?
Note that what I am proposing is an abstraction of, say, the group $G$ of permutations over $\mathbb N$ or the group $H$ of permuations over $\mathbb Q$. Much like how ordinals are the canonical representatives of well-orders of a certain size, the groups I'm discussing would be the canonical representatives of the permuation group over sets of a certain size, (e.g., $G$ and $H$ above would both be isomorphic to the group $S_{\aleph_0}$).
Part of the reason I ask this is that I have no idea if anything I said even makes sense, so I appreciate any and all thoughts on this matter. Thanks!
This is actually a heavily studied subject. Am example reference:
Peter Cameron, Permutation Groups, 1999
(most of the book is on finite permutation groups).