Show that every group $G$ is isomorphic to a permutation group Cayley's Theorem states that "every group $G$ is isomorphic to a permutation group". But in every book we only get it for finite groups. Can anyone please help me to solve this for every group.
 A: Every group is isomorphic to a permutation group, irrespective of cardinality. The only difference is how to define the appropriate symmetric group to embed the group into. If $G$ is finite then the standard idea is to embed $G$ into the symmetric group $S_{n}$ where $n=|G|$. We can work more generally by defining $\operatorname{Sym}(G)$ to be the set of bijections from the set $G$ to $G$ (so permutations on the set underlying $G$). If $G$ is finite then there is a clear bijection between $S_{|G|}$ and $\operatorname{Sym}(G)$.
The rest of the proof is identical.
I suspect the reason that introductory textbooks only prove Cayley's theorem for finite groups is because they are focusing on finite permutation groups. Infinite permutation groups are rather different, and it is sensible to avoid them in a first text. For example, there is no analogue of the alternating group for infinite symmetric groups, although a relatively easy tweak in the definition of these symmetric group does give us infinite alternating groups.
