Sum of elements in row of character table is positive integer. If $G$ is a (finite) group, how can I prove that in the corresponding character table, the sum of the elements in any row is a non-negative integer? The hint in the book says that I should let $G$ act on $G$ by conjugation and then consider the permutation character. I really don't understand this hint, can someone help?
 A: Let $U,V$ be the representation of the group $G$. Then 
$$(\chi_U,\chi_V)=\dim_{\mathbb{C}}\mathrm{Hom}_{\mathbb{C}[G]}(U,V)$$
is a non-negative integer. This is a theorem maybe in your textbook. By definition
$$(\chi_U,\chi_V)=\frac{1}{|G|}\sum_{g\in G}\chi_U(g)\overline{\chi_V(g)}=\frac{1}{|G|}\sum_{t=1}^rk_t\chi_U(g_t)\overline{\chi_V(g_t)},$$
where $r$ is the number of the conjugate class in $G$，$k_t$ is the number of elements in $t$th conjugate class, that is $k_t=[G:C_G(g_t)]$ and $g_t$ is the representation element of the conjugate class.
Key: If we can choose some special representation $V$ of the group $G$ such that
$$k_t\overline{\chi_V(g_t)}=|G|,~~~~~\forall 1\leq t\leq r~~~~~(*)$$
The proof will be done.
The hint tells you what is the special representation $V$ of the group $G$. 
"Let $G$ acts on $G$-set $X:=G$ by conjugation
$$g(x):=gxg^{-1},~~~~~\forall g\in G,~~~\forall x\in X,$$
and then regard it as the permutation representation $V$ of the group $G$" gives you these two conditions:


*

*$$\chi_V(g_t)=\mathrm{tr}\big(\rho(g_t)\big)=|\{x\in X:x=g_t(x)=g_txg_t^{-1}\}|=|\{x\in X:xg_tx^{-1}=g_t\}|=|X_{g_t}|.$$
We regard $xg_tx^{-1}=x(g_t)$ as $X$ acts on $G$ by conjugation we talk before and $X_{g_t}$ is the stabilizer of $g_t$ under the action. Obviously, $\chi_V(g_t)$ is an integer.

*The number of the orbit of $g_t$ under the action we talk above is exactly 
$$|X(g_t)|=k_t=[G:C_G(g_t)].$$ 
Now, $(*)$ follows by the "Orbit-Stabilizer Fomular".
