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



Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$
This appears to be well-studied, see e.g. <http://www-bcf.usc.edu/~fulman/LAApaper.pdf>
and references therein.


