How are proofs in group theory involving group actions valid when we choose a particular group action? Doesn't that involve a loss of generality? For example, I am looking at a proof of Cayley's theorem that uses the action of a group on itself by left translation. I understand the proof, but have a hard time seeing why we're working with a specific group action, because doing that makes it seem like we're using some trickery and there are a ton of group actions to choose from so why are we choosing left translation and not, say, conjugation?
 A: Not every action will work, and in particular conjugation will not work if $Z(G)$ is non-trivial. We need to pick an action with a specific property, and left translation just so happens to have this property. Right translation will also work, for example.
For an action to work in the proof of Cayley's theorem you need that every element moves something in the action, which is called a faithful action. That is, for all $g\in G$ we need an $x\in G$ such that $g\cdot x\neq x$ (here, I am acting by $g$). Otherwise, if $g$ fixes everything then it is contained in the kernel of the action and our map $G\rightarrow S_{|G|}$ is not an injection. Left translation ensures that every non-trivial element moves something. However, conjugation can have fixed points (every element of $Z(G)$).
A: Why wouldn't such proofs be valid? After all the Cayley's theorem is not about any group action. The particular group action is only used as a tool to prove it.
Such choices are all over maths. For example how often do you use natural numbers as a particularly important set? Or the sphere $S^1$, so important in algebraic geometry? And so on, and so on...
