Considering transitive $G$-set 
Question. Suppose that $X$ is a transitive $G$-set of size greater than $1$ and let $\pi$ be the associated permutation representation with the character $\chi$. Show that some element $g \in G$ has no fixed points.
The start of the solution goes: Suppose all $g \in G$ has fixed points, so $\chi(g) \geq 1$. Then define $\chi_2=\chi-\chi_{\mathbb{I}}$, so $\chi_2(g) \geq 0$ for all $g \in G$. $\chi_2$ is also a character and $\langle \mathbb{I},\chi_2 \rangle=0$ since the action is transitive. Now
$$0=\langle \mathbb{I},\chi_2 \rangle=\frac{1}{|G|}\sum_{g \in G}\chi_2(g) >0$$
because all terms are non-negative and the one for $g=1$ is $\dim \chi_2 >0$.

I cannot see why $\chi_2$ is also a character.
I cannot see why $\langle \mathbb{I},\chi_2 \rangle=0$ because the action is transitive.
Nor can I see why $\dim \chi_2 >0$.
 A: Notice that $[\chi,\mathbb{I}]=k$ is equal to number of the orbits.. Since our action is transitive  $k=1$.
$$\chi=\mathbb{I}+n_2\phi_2+ n_3\phi_3...$$
where $\phi_i$ are irreducible character of $G$. Form this, it is easy to see that why $\chi_2$ is an character. 
Notice also that if $\dim(\chi_2)=0$ then $\chi_2=0\implies \chi=\mathbb{I}$ hence $X$ has only one elements which is a contradiction.
EDIT: $$\chi=\mathbb{I}+n_2\phi_2+... n_k\phi$$
where $n_i$ are nonnegative integers and $\phi_i$ are irreducible characters.
$$\chi_2=\chi-\mathbb{I}=n_2\phi_2+...+n_k\phi_k$$
$$\dim(\chi_2)=\chi_2(e)=n_2\phi_2(e)+...n_k\phi_k(e)$$
if $0=\dim(\chi_2)$ then we must have $n_i=0$ for $i\geq 2$ which means $\chi=\mathbb{I}\implies$ $G$ fixes every elemets of $X$ which is not the case as $|X|>1$ and the action is transitive.
Notice that if a function is a linear combination of irreducible characters with non-negative integer coefficient (and at least one of them is positive) then the function is a character. Thus, $\chi_2$ is an character.
Edit $2$: Having $p:G\to Gl_n(\mathbb C)$ is equivalent that you have an $n$ dimensional $\mathbb C G$ module $V$ and vice versa.
Existence of $\phi_i$ (irreducable character) means that we have irreducable $\mathbb C G$ module $V_i$ and corresponding represantation  $p_i:G\to Gl_{n_i}(\mathbb C)$.
A natural question is that can we find representation for $V_i\oplus V_j$ ?
\begin{pmatrix}
  p_i & \huge 0  \\
  \huge 0 & p_j  \\
  \end{pmatrix}
 You can see that above is the representatin of $V_i\oplus V_j$. And the corresponding character is exactly $Tr(p_i)+Tr(p_j)=\phi_i+\phi_j$." By similiar reasoning $n\phi_i+m\phi_j$ is the character of the module 
$$V_i\oplus ...V_i\oplus V_j\oplus....\oplus V_j$$
Where $V_i$ is summed $n$ times and $V_j$ is summed by $m$ times. (Of course $m,n$ are nonnegative integer and at least one of them is positive)
By above argument, you can see that linear combinations of characters by non-integers (at least one of them is positive) is a character.
