Can Cayley's Theorem be applied to groups with infinite order?

Question

I've been reading Joseph Gallian's Contemporary Abstract Algebra and I'm reading about isomorphisms. I came across Cayley's Theorem, which states:

Every group is isomorphic to a group of permutations.

I asked my professor if this is true for groups with infinite order, like $\mathbb{Z}$, and he said that Cayley's Theorem only applies to groups with finite order. However, after looking at the proof of Cayley's Theorem in the book, it seems that it doesn't have to be constrained to finite groups. Is there anyway to be crafty and come up with such a group of permutations that is isomorphic to, say, $\mathbb{Z}$ under addition? If not, why?

Answer 1

The group $\mathbb Z$ is isomorphic to the group of all permutations of $\mathbb Z$ of the form $m\mapsto m+n$ ($n\in\mathbb Z$).

More generally, if $G$ is any group, then $G$ is isomorphic to the group of all permutations of $G$ of the form $g\mapsto gh$ ($h\in G$).

Answer 2

Your professor is incorrect. Any group $G$ acts on itself by left multiplication, and this action gives an embedding of $G$ in $\operatorname{Sym}(G)$.

Thus every group is isomorphic to a subgroup of a group of permutations. And that is precisely what Cayley's theorem states.