I need an example of a function with these properties. I have a problem that says $A$ is finite and $B\subset A$ and that $G$ is the subset set of $S_A$ consisting of all the permutations $f$ of $A$ s.t. $f(x)\in B \ \forall \ x\in B$. 
these functions are bijections on B because it is finite, and we have that 1-1 gives us onto, but I am trying to think of an example where if we didnt have that A was finite, then we might not necessarily have that they are bijections on B.
Can someone think of a good example?

To give a simple example, let $A = \mathbb Q$, $B = \mathbb Z$ and $f\colon\mathbb Q \to \mathbb Q$, $x \mapsto 2x$. Then $f[\mathbb Z] = 2\mathbb Z \subsetneq \mathbb Z$.
