Proving that RSA encryption function with non-square free modulus is not a permutation
While most of the answers boil down to efficency and security considerations, there is a single answer which states that RSA encryption function with modulus consisting of prime powers loses it's bijection properties, i.e, it is not a permutation any more. However this behaviour is shown only on example with no proof.

It's rather straightforward to demonstrate (assuming $e&gt;1$; with $e=1$, it is a permutation, but not a very interesting one).
A value $N$ is nonsquarefree if there is a value $p&gt;1, q$ such that $N = p^2q$ (note that $q$ may have $p$ as a factor).  If so, then consider the encryption of the two values $0$ and $pq$.  In the two cases, we have:
$$0^e \equiv 0 \pmod N$$
$$(pq)^e \equiv p^eq^e \equiv p^{2+x}q^{1+y} \pmod N$$
for $x = e-2$ and $y = e-1$.  Now, both $x, y \ge 0$, and so $p^{2+x}q^{1+y}$ is a multiple of $p^2q$, and so this latter is equivalent to $0 \bmod N$
Since these two distinct plaintexts map to the same ciphertext 0, the mapping cannot be bijective.