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**Drexel28** I'm a little bit confused, about the hint. What you're basically saying is that if $\displaystyle \chi\in\text{irr}(G)$ then $\displaystyle \displaystyle \sum_{g\in G}\chi(g)\in\mathbb{N}\cup\{0\}$, right? But, think about it if we let $\displaystyle \chi^{\text{triv}}$ denote the trivial character $\displaystyle g\mapsto 1\in\mathbb{C}$ then by the orthogonality relations for irreducible characters we have that $\displaystyle \displaystyle \left\langle \chi,\chi^{\text{triv}}\right\rangle=\delta_{\chi, \chi^{\text{triv}}$. But, by definition $\displaystyle \displaystyle \left\langle\chi,\chi^{\text{triv}}\right\rangle=\ frac{1}{|G|}\sum_{g\in G}\chi(g)\overline{\chi^{\text{triv}}(g)}=\frac{1} {|G|}\sum_{g\in G}\chi(g)$ so that $\displaystyle \displaystyle \sum_{g\in G}\chi(g)=\delta_{\chi,\chi^{\text{triv}}}|G|$, which is what we want, right?