(I asked the following question at StackExchange but received no answer.)

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.

I want to show that $D=\mathrm{Rep}(G/Z(G))$.

It seems that this problem is claimed in Tensor Categories By Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik without a complete proof.

I appreciate any help.

(I asked the following question at StackExchange but received no answer.)

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \otimes V^*$ for each (representative of isomorphism class of) simple object of $C$.

I want to show that $D=\mathrm{Rep}(G/Z(G))$.

It seems that this problem is claimed in Tensor Categories By Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik without a complete proof.

I appreciate any help.