# Exercise 4.2, I. Martin Isaacs' Character Theory

Problem : Let $\mathcal{K}$ be a conjugacy class in a finite group $G$ which is not contained in any proper normal subgroup of $G$. Let $K$ be the corresponding class sum in $\mathbb{C}[G]$ and let $m$ be the number of distinct values of $w_\chi(K)$ for $\chi\in Irr(G)$. Recall that $w_\chi(K)=\frac{\chi(g)|\mathcal{K}|}{\chi(1)}$. Show that every element in $G$ can be written as a product of fewer than $m$ elements of $\mathcal{K}$.

In the hint, the author said that Mimic the proof of Theorem 4.3 using the second orthogonality relation but I have not still found a proof. In fact, mimic the proof of Theorem 4.3, I proceed as follows : let $a_1,\cdots,a_m$ be $m$ distinct values of the $w_\chi(K)$. Let $G_i=\{\chi\in Irr(G) : w_\chi(K)=a_i\}$. Let $x\in G$. If $x\in\cal{K}$ then it is done. Otherwise, $x$ is not conjugate with any element $g\in\cal{K}$. Using the second orthogonality relation, we have that

$0=\sum_{\chi\in Irr(G)}\chi(g)\overline{\chi(x)}=\sum_{\chi}\frac{w_\chi(K)\chi(1)\overline{\chi(x)}}{|\cal{K}|}=\frac{1}{|\cal{K}|}\sum_{i=1}^ma_i\sum_{\chi\in G_i}\chi(1)\overline{\chi(x)}$

But then I dont know how to go on. Could you give me a hint. Thanks in advance.

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