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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.

Thanks a lot in advance.

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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.

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  • $\begingroup$ Ok I think I got it! $\endgroup$ – Rijubrata Oct 15 '18 at 0:26

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