Show that the symmetric group Sn is nonabelian for n geq 3.

Consider first the case where n=3 along with the following transpositions:
$\psi =(12)and\sigma =(23)$
Notice that $\psi \times \sigma =(123)$ while $\sigma \times \psi =(132)$, which shows that $\psi \times \sigma$ is not equal to $\sigma \times \psi$ i.e. S3 is not abelian.
Now, these two permuations are in every single symmetric group of order $\ge 3$. Therefore we must conclude that Sn is not abelian for $n\ge 3$