# Understanding conjugacy classes in $SL_{2}(\mathbb{F}_{q})$

I am trying to calculate the conjugacy classes of the group $$SL_{2}(\mathbb{F}_{q})$$, with the help of the knowledge of conjugacy classes of $$GL_{2}(\mathbb{F}_{q})$$.

I am using two of the following results:

1) Let $$G$$ be a group and $$H$$ a normal subgroup of $$G$$. Suppose $$h\in H$$. It is clear that $$Cl_{G}(h)\subset H$$. Then $$Cl_{G}(h)$$ splits into equal portions in $$H$$ and the number of parts into which it splits is $$[G:HC_{G}(h)]$$, where $$C_{G}(h)$$ is the centraliser of $$h$$ in $$G$$.

One can easily calculate the representatives of the split classes using the above result if one proves this result, which I have done.

2) This result in particular involves the two groups in the question in particular, and also gives knowledge about the splitting. This goes as follows:

Consider the map $$det: C_{G}(h)\to \mathbb{F}_{q}^{*}$$ given by the obvious determinant map. Now, call the image of this map as $$L$$. Then again the number of classes into which $$Cl_{G}(h)$$ splits is given by the index $$[\mathbb{F}_{q}^{*}:L]$$.

Again if one proves this theorem then one can also find the representatives of the split classes.

Now, with these two theorems in hand, and also the fact that I know the conjugacy classes of $$GL_{2}(\mathbb{F}_{q})$$, representatives of those classes and the centralisers of each such representative elements, I have managed to understand the splitting of each classes in $$SL_{2}(\mathbb{F}_{q})$$ in all but one case. The following is the case which I don’t understand:

Consider the conjugacy class of the matrices whose characterestic polynomial is given by an irreducible polynomial of degree 2 over $$\mathbb{F}_{q}$$. I have found representatives of such classes, which looks as follows $$M= \left[ {\begin{array}{cc} x & \epsilon y \\ y & x \\ \end{array} } \right]$$

Where $$y\neq 0$$ and $$\epsilon$$ is a non-square element in the field. It’s centraliser is

$$\{ \left[ {\begin{array}{cc} x & \epsilon y \\ y & x \\ \end{array} } \right] \}$$, $$x,y$$ both not zero.

But using this information I have failed to apply the results 1 or 2, to understand whether this class splits as conjugacy classes of $$SL_{2}(\mathbb{F}_{q})$$.

So, I ask how can I solve this problem. Also a general kind of result will be nice because in that case of I can also understand the splitting in $$SL_{3}(\mathbb{F}_{q})$$ of these semi-simple classes, whose characterestic polynomial doesn’t have all roots in the base field.

Your degree 2 case is relatively easy as it simply amounts to whether the binary quadratic form $$x^2-\epsilon y^2$$ represents all elements of $$\mathbb{F}_q^*$$. However, $$x^2-\epsilon y^2$$ is basically the norm map $$\mathbb{F}_{q^2}\to\mathbb{F}_q$$ and thus is surjective.
This also generalizes to higher degree $$n$$:- if a representative has at least one semisimple Jordan block then the determinant map is basically the norm map and so is surjective. For example, for $$n=3$$ the only possible splitting to investigate is with the minimal polynomial $$(t-\alpha)^3$$ case.