[GAP Forum] Questions about representations in Sn and Tz Transformations

[Alexander Hulpke]

Dear Nikos Apostolakis, Dear Forum,

> I am looking for representations of some fp groups in to symmetric
> groups.  Searching the documentation and the archives I came up with
> the following code:
>
>
> My first question is: Is there a way to specify *a priory* that the
> generators will be mapped within a given conjugacy class?  For
> example I would like all the generators to be mapped to
> transpositions.  (I work with Wirtinger presentations of knot and
> link groups where the generators are in relatively few conjugacy
> classes.)

Using slightly different code (the machinery driving `GQuotients'  
this would be possible. However this extra testing comes at an extra  
cost in runtime. If your image group is (likely to be) Sn, and if  
your presentation contains a relator a^2=1 (if the image of a is to  
be an involution we can easily demand this, if its not yet in the  
presentation) my guess is that `LowIndexSubgroups' actually will run  
faster.

>
> The second question has to do with Tietze transformations.  The
> above process seems to be faster if I first apply the function TzGo
> to the presentation.  Also checking if the resulting representation
> maps in the desired conjugacy cass is easier since there are fewer
> generators.  However, I need to know the images of the original
> generators.  So the question is, after I apply TzGo how can I get
> the old generators expressed in terms of the new ones?  I mean,
> by setting "printLevel=2" I can have this information printed out
> but how can I use it in a program?

If you use `IsomorphismSimplifiedFpGroup' instead of running the low  
level Tietze code, you get a nice homomorphism that gives you all  
translation information.

Best,

    Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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