I have the following question: Consider the following two finite groups: $GL_{n}(q)$ and $SL_{n}(q)$. What I am trying to understand is the regular semisimple conjugacy classes of $SL_{n}(q)$. Now, from the theory of canonical forms, one can easily find the regular semisimple conjugacy classes in case of $GL_{n}(q)$. From there, I am trying to figure out what would be the case in $SL_{n}(q)$. We know if $u\in SL_{n}(q)$ is regular semisimple, then the $GL_{n}$ conjugacy class of $u$ is contained in $SL_{n}$, but it can split in $SL_{n}$. If it splits, then it will split into equal parts. So, basically I am trying to find out if such a class will split and under what conditions.

I have tried this for $SL_{3}(q)$, and by using some elementary techniques of finite group theory, I have come to the conclusion, that the $GL_{3}$ semisimple regular classes in $SL_{3}$ , doesn't split. I want to know, whether such a thing is true in general.

I Know some theory of algebraic groups over algebraically closed field, but I am not aware of how those result are carried over to finite fields, though I think that such things might be relevant in this case. I hope one can give a rough idea of the problem or point out where to look for getting an answer.

Thank you