Why is the symmetric group abelian only when its order is less than or equal to two?

Question

I was reading online about the topic and this theorem came out, but I haven't found any proof for it

Answer 1

because $(1,2)(2,3)=(2,3,1)$ and $(2,3)(1,2)=(1,3,2)$

Answer 2

These proofs and this question show that the center of a symmetric group is trivial, as in it only contains the identity element. It follows that the group will always be nonabelian since the center will never equal the group (which is required for abelian groups) ...except, of course, $S_2$. Every element in that group commutes with every other element.