Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 8 / 5 / i / Solution

Take $H={\mathbb{C}}^{S_3}$, the function Hopf algebra of a finite group. Multiplication is pointwise, andThe group associativity, identity and inverse laws respectively verify coassociativity, the counit and the antipode; pointwise products show that $\Delta ,\epsilon$ are algebra homomorphisms. Thus $H$ is a commutative Hopf algebra.

For $g=(12)$ and $h=(23)$, using right-to-left permutation composition, $gh=(123)$ whereas $hg=(132)$. The delta function $f={\delta }_{(123)}$ has $(\Delta f)(g,h)=1$ but $({\Delta }^{\mathrm{op}}f)(g,h)=0$. Therefore $\boxed{{\mathbb{C}}^{S_3}\text{ is commutative but noncocommutative}}$.