Reading this problem I remembered trying to solve the following problem. For a set $A$, denote by $S_A=\{ f : A \to A | f \text{ is bijective }\}$. Denote by $|X|$ the cardinal number of $|X|$.
Prove that for two sets $X,Y$ we have $|X|=|Y| \Leftrightarrow |S_X|=|S_Y|$.
I didn't manage to solve the case where $X,Y$ are infinite, and don't even know how to start. I tried to represent $X$ as a subset of $Y$( if $|X|<|Y|$) and maybe find a permutation in $S_Y \setminus S_X$.