Number of conjugacy classes are there over $\mathbb{F}_p$ I am struggling with the next exercise of my HW:
How many conjugacy classes are in $GL_3(\mathbb{F}_p)$? And how many in $SL_2(\mathbb{F}_p)$?
It's on the topic of Frobenius normal form of finitely generated modules over $\mathbb{F}_p$.
I'd appreciate any idea.

For the general linear group I suggest that you count the (irreducible) linear polynomials with non-zero constant term, the irreducible quadratics, and finally the irreducible cubics. You then get a conjugacy class for each cubic, $C(f(X)$; a conjugacy class for each pair (linear, irreducible quadratic), $C(X-\alpha)\oplus C(q(X)$; and then you are left dealing with the elements where the characteristic polynomial is the product of linear factors. These last will give you classes for types $C((X-\alpha)^3)$, $C((X-\alpha)^2)\oplus C(X-\beta)$, $C(X-\alpha)\oplus C(X-\beta) \oplus C(X-\gamma)$. (Note, $\alpha=\beta$ is possible.)
For the special group you now need to identify which of these classes is in the group, and then investigate the relative sizes of the centraliser of an element in the special group and the general group to see whether the $GL$-orbit splits into smaller $SL$-orbits.   
