> < ^ Date: Thu, 27 Aug 1998 15:48:37 +0100
< ^ From: Daniella Bak <dbak@email.GC.cuny.edu >
< ^ Subject: Re: conjugates and conjugacy classes

Thank you so much. I didn't even check into this yet, but it looks great.

Frank Luebeck wrote:

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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