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

Show that the symmetric group Sn is nonabelian for $n\ge 3$.

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Consider first the case where n=3 along with the following transpositions:

Notice that $\psi ×\sigma =\left(123\right)$ while $\sigma ×\psi =\left(132\right)$, which shows that $\psi ×\sigma$ is not equal to $\sigma ×\psi$ 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 $n\ge 3$