I think the author accidentally described the dual of the Hopf algebra you're thinking of. 
Finite group rings are usually endowed with multiplication $(g,h)\mapsto gh$ and comultiplication $g \mapsto g\otimes g$ (see [here][1]). 

The coordinate ring  $k[G]$ is obtained by dualizing. Then $g \mapsto g\otimes g$ becomes 
$e_g^2 = e_g$, where $e_g$ is the function on $G$ that maps $g$ to $1$ and all other group elements to $0$.
 

  [1]: http://en.wikipedia.org/wiki/Group_Hopf_algebra