What are the conjugacy classes of $PSL (3,q)$ and $PSU(3,q)$?



Conjugacy classes in PSL(3,q) and PSU(3,q)
This is well-known, and there are a number of relevant references. Firstly, there are these by Wall (they are pretty hard to read though).



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> Wall, G. E. Conjugacy classes in projective and special linear groups. Bull. Austral. Math. Soc. 22 (1980), no. 3, 339–364.
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> Wall, G. E. On the conjugacy classes in the unitary, symplectic and orthogonal groups.
> J. Austral. Math. Soc. 3 1963 1–62.
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You might also be interested in this paper which I personally find much more readable.



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> Macdonald, I. G.
> Numbers of conjugacy classes in some finite classical groups.
> Bull. Austral. Math. Soc. 23 (1981), no. 1, 23–48.
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You could also look at Carter's *Finite groups of Lie type* (email me if you want a copy) although that is a rather hard text and is much more general than you need.


For conjugacy in $PSL\_3(q)$ you could also just do a search on *Jordan rational forms* - these forms classify conjugacy in $GL\_3(q)$, and this classification can then be adapted to deal with $PSL\_3(q)$. [I have a paper with Anupam Singh](http://arxiv.org/abs/0809.4412) in which we describe how to do this (again in much greater generality than you need).


