# Splitting of regular semisimple conjugacy classes in $SL_{n}(q)$

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

• does www-bcf.usc.edu/~fulman/LAApaper.pdf contain answers to your question? Feb 4, 2019 at 3:40
• references in the link I just cited come handy too. It seems a lot is known about your question. Feb 4, 2019 at 3:41