Let $A$ be an infinite set and let $S$ be the set of all bijections $A \rightarrow A$. Then if $\mid A \mid = \kappa$, then $\mid S \mid = 2^\kappa$.
I'm able to prove it for $A = \mathbb{N}$ by showing an injection $P(\mathbb{N}) \rightarrow S$, but how can I prove it for any set $A$?
It suffices to exhibit an injection $P(A) \to S$.
Let $X$ be a subset of $A$, and fix a derangement $d_X:A \setminus X \to A \setminus X$ (see this question for proof of existence of $d_X$). Now define the bijection $f_X:A \to A$ as follows:
$f(x)= \begin{cases} x, & x \in X \\[2ex] d_X(x), &x \notin X \end{cases} $
The map $X \mapsto f_X$ is our required injection.