Let $n\ge 3$. Then the dihedral group on $n$ vertices, $D_{2n}$, is not abelian.

Proof

The most natural dihedral group presentation is $⟨a,b\mid a^n,b^2,ba{b}^{-1}=a^{-1}⟩$. In particular, $ba=a^{-1}b=a^{-2}ab$, so $ab=ba$ if and only if $a^2$ is the identity. But $a$ is the rotation which has order $n>2$, so $ab$ cannot be equal to $ba$.