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On Tue, 25 Aug 1998, Daniella Bak wrote:

I'm a new user of GAP, and I realize that this may be an elementary

question. I was wondering if someone could tell me how to find conjugate

elements in (and/or conjugacy classes for) PSL(2,q) as well as in the

braid groups (B3,B4 etc.)

On Tue, 26 Aug 1998, she made the question more precise:

With regards to PSL(2,q), I want to look specifically at q=5.

With regards to both this and the braid group (B4,B5 - the infinite group

is not as important to me), I really want to know about representatives of

the conjugacy classes, as well as their orders.

Dear Daniella,

For small q (e.g., q=5) the brute force method can handle PSL(2,q)

(this essentially uses translations to permutation groups) - try the

following commands:

sl := SL(2,5); z := Centre(sl); z.name := "z"; # just for nicer printing psl := sl/z; cl := ConjugacyClasses(psl);; rep := List(cl, Representative); ord := List(rep, x-> Order(psl, x));

(Of course, in this case you could also use the permutation

group a5 := Group((1,2,3,4,5), (1,2,3)); which is isomorphic to psl.)

It is also not difficult to write down explicit representatives for

the conjugacy classes of PSL(2,q) (as matrices in SL(2,q)) for general q

essentially using Jordan normal form of matrices.

Concerning the "braid groups" you mention I'm still not sure what you

mean.

If you mean the abstract fintely presented groups which I know as "braid

groups" or "Artin-Tits braid groups" the general problem of finding

conjugacy classes of these infinite groups is not yet solved.

Or do you mean by "B4", "B5" the finite Weyl groups of this type?

These can be created with the command 'CoxeterGroup' from the CHEVIE

package and their conjugacy classes are found by 'ConjugacyClasses'.

RequirePackage("chevie");

b5 := CoxeterGroup("B", 5);

cl := ConjugacyClasses(b5);

I hope this helps,

Frank

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