Let $G$ be a group and $C_{1},C_{2},\dotsc ,C_{s}$ be conjugacy classes of $G$ and $K_{1}, K_{2},\dotsc , K_{s}$ be the class sums in $\mathbb{C}[G]$. Suppose there exists $c \in \mathbb{C}$ such that $\sum K_{i} = c\prod K{i}$. Show that $G=G'$. Conversely if $G'=G$ then show that there exists $c\in \mathbb{Q}$ such that $\sum K_{i} = c\prod K{i}$.
This is a problem from Isaacs' book of character theory. A hint is given as follows: if $\chi$ is not trivial representation and $\chi$ is irreducible then show that $\frac{h_{i}\chi(g_{i})}{\chi(1)}=0$ for some $i$, where $g_{i}$ is a class representative of $K_{i}$ and $h_{i}$ is the size of the conjugacy class $C_{i}$. I need help in this problem. I have neither been able to show the hint nor have any idea what the hint will suggest.
Thanks in advance!!!