Comultiplication in Convolution algebra: Where is the mistake?

Question

Let $G$ be a finite group and $C(G)$ its convolution Hopf algebra. It is known that the characters $\widehat{G}$ form an orthogonal basis in $C(G)$ and $\sum_{\chi \in \widehat{G}}\, \chi = |G|\cdot \delta_e$, where the delta function $\delta_e$ is the unit in $C(G)$. Moreover, the characters are grouplike elements. If this is all true, what am I doing wrong in the following chain of equations:$$ \sum_{\chi} \chi \otimes \chi =\sum_\chi \Delta(\chi) = \Delta\left(\sum_\chi \chi\right) =|G|\Delta(\delta_e)=|G|\delta_e \otimes \delta_e=\frac{1}{|G|}\sum_{\chi,\chi'} \chi\otimes \chi',$$ which can't be since $\chi\otimes \chi'$ form a basis of $C(G)\otimes C(G)$?!

Answer 1

The mistake lies in your assertion that the characters are grouplikes. In fact, if one considers the characters to be grouplikes, applying the comultiplication map $\Delta:C(G)\rightarrow C(G)\otimes C(G)$, to both sides of the relation $$ \sum_{\chi \in \widehat{G}}\, \chi = |G|\cdot \delta_e $$ easily yields a contradiction.