Function Hopf algebra of a finite group
= Function Hopf algebra of a finite group
{title2=$k^G$}
Functions on a finite group form a <Hopf algebra> under pointwise multiplication, with $(\Delta f)(g,h)=f(gh)$, $\varepsilon(f)=f(1)$ and $(Sf)(g)=f(g^{-1})$. The coproduct is cocommutative exactly when the group is abelian, since delta functions distinguish distinct products $gh,hg$.