Fulton and Harris exercise 5.7 I would like to know if I'm on the right track:
We're trying to derive the character table for $PGL(2,q=p^k)$ from $GL(2,q)$. The table for $GL$ is given in the text.
It gives the hint that the characters of $PGL$ are those of $GL$ that take on the same values for elements equivalent mod $\mathbb{F}_q^*$; that is to say, matrices equivalent mod scalar multiplication.
So, for instance, this is one kind of representation of $GL$:
$U_{\alpha}: \alpha(x^2); \alpha(x^2); \alpha(xy); \alpha([x+y\sqrt{\epsilon}]^q)$
(Each over a different conjugacy class, I've skipped the particular detail here.)
So for instance I want to find $\alpha$ satisfying $\alpha(k^2x^2)=\alpha(x^2)$, $\alpha(k^2xy)=\alpha(xy)$, $\alpha(k[x+y\sqrt{\epsilon}]^q)=\alpha([x+y\sqrt{\epsilon}]^q)$ (each representing multiplying a matrix by a scalar).
Is that right? Is it productive? It's been tedious and I don't know where I'm going for the more difficult representations.
 A: The basic idea is that all representations of a quotient group come from an original group. 
Since you are given the character table of $GL(2, q)$, all you have to do is determine which give the irreducible characters of $PGL(2, q)$. The following fact is useful.
$$ \ker \rho_V = \ker \chi_V := \{\, g \in G \mid \chi_V(g) = \chi_V(1) \,\} $$
Since the kernel of the desired characters contain the center $Z(GL(2, q)) = \mathbb{F}_q^*I =: N$, you can find out the desired characters $\chi_V$ from these data. Then $\psi_V(gN) = \chi_V(g)$ define irreducible characters of $PGL(2, q)$. (You can check that by orthogonality relations or prove it in general.)
As a function, irreducible characters can be determined as described above. But this construction doesn't provide information about conjugacy classes (although you can guess from the values and the number of irreducible characters). So you have to check the guess by examining each ones.
%% Since this principle works in any case, it would be better to see how it works in a simple case before you get into cumbersome calculations if you are not familiar with. (E.g., $G = S_3$, $N=A_3$.)
