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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

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This appears to be well-studied, see e.g. http://www-bcf.usc.edu/~fulman/LAApaper.pdf and references therein.

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    $\begingroup$ Maybe you can extract from section 2 of the paper an explicit answer to the question asked? Anyway you are correct about the question being well-studied. $\endgroup$ – Jim Humphreys Feb 4 at 21:38
  • $\begingroup$ Thanks for the reference! $\endgroup$ – Rijubrata Feb 5 at 7:51

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