We write $[x,y]$ for the commutator $x^{-1}y^{-1}xy$ of $x$ and $y$ in a group $G$.

(A) Let $g \in G$ and fix $x \in G$. Show that $g$ is conjugate to $[x,y]$ for some $y \in G$ iff

$$\sum_{\chi \in \text{Irr}(G)} \frac{|\chi(x)|^{2}\bar{\chi(g)}}{\chi(1)} \neq 0.$$

(B) Show that if $g=[x,y]$ for some $x,y \in G$ iff

$$\sum_{\chi \in \text{Irr}(G)} \frac{\chi(g)}{\chi(1)} \neq 0.$$

I have done the first part. I think that second part is just applying the first part in a correct way but I cannot do it. This problem is important in the sense that it determines all the commutators of a group from the character table. The set of commutator generally does not form a subgroup. Any hint or solution will be appreciated. Thanks in advance.

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