# Conjugacy classes in PSL(3,q) and PSU(3,q)

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

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## 3 Answers

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).

Wall, G. E. Conjugacy classes in projective and special linear groups. Bull. Austral. Math. Soc. 22 (1980), no. 3, 339–364.

Wall, G. E. On the conjugacy classes in the unitary, symplectic and orthogonal groups. J. Austral. Math. Soc. 3 1963 1–62.

You might also be interested in this paper which I personally find much more readable.

Macdonald, I. G. Numbers of conjugacy classes in some finite classical groups. Bull. Austral. Math. Soc. 23 (1981), no. 1, 23–48.

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 in which we describe how to do this (again in much greater generality than you need).

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As Nick points out, the classes are well documented in the literature. An old paper by Frame and Simpson, with free online access here, is a convenient source. (This paper was in fact reviewed by G.E. Wall.) Their combined treatment of the groups leads them to mix up the parametrizations a bit in the character tables, but I think the class information is basically reliable. In any case, the later work on characters inspired by Deligne and Lusztig (as in Carter's 1985 book) has led to a far more thorough treatment, while the cpnjugacy class information has been developed uniformly for all finite groups of Lie type. Lots of literature out there.

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The computer algebra system GAP implements the recursion formulas for the numbers of conjugacy classes from Macdonald's paper cited in Nick Gill's answer. This covers the groups ${\rm GL}(n,q)$, ${\rm GU}(n,q)$, ${\rm SL}(n,q)$, ${\rm SU}(n,q)$, ${\rm PGL}(n,q)$, ${\rm PGU}(n,q)$, ${\rm PSL}(n,q)$ and ${\rm PSU}(n,q)$ for positive integers $n$ and prime powers $q$.

Some examples:

gap> List([2,3,4,5,7,8,9,11,13,16],q->NrConjugacyClassesPSL(3,q));
[ 6, 12, 10, 30, 22, 72, 90, 132, 64, 94 ]
gap> List([2,3,4,5,7,8,9,11,13,16],q->NrConjugacyClassesPSU(3,q));
[ 6, 14, 22, 14, 58, 28, 92, 48, 184, 274 ]
gap> NrConjugacyClassesPSL(3,2^128);
38597363079105398474523661669562635951203422344187167500973652871781632617134
gap> NrConjugacyClassesPSU(3,NextPrimeInt(10^15));
333333333333358333333333333806
gap> NrConjugacyClassesPSL(48,43);
10117782215999949055454469829685675266925185945227906522217956917437261029288
gap> NrConjugacyClassesPSU(45,49);
47750910568826421436319984311888622181462015639973663869365087657134297408

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