[GAP Forum] transfer fp relators

[Rudolf Zlabinger]

Dear Forum,

I am looking for a possibility to transfer relators of one fp group to another free group without transcripting the relators manually. I used in the following example a method, which is very restricting for the variable name of the target free group (= f) and is not allowed in a function, as AssignGeneratorVariables is forbidden there. Is there another possibility?

As an example I reproduced the direct product of symmetric group (3) and alternating group (4). The manual method to produce the mutual relators rel0010 - rel0040 is no problem to me, as this is normally done by automatic methods in my functions.

gap> f:=FreeGroup(4);;
gap> AssignGeneratorVariables(f); 
#I  Assigned the global variables [ f1, f2, f3, f4 ]
gap> s3:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> a4:=AlternatingGroup(4);;
gap> s3fp:=Image(IsomorphismFpGroup(s3));;
gap> a4fp:=Image(IsomorphismFpGroup(a4));;
gap> prod:=FreeProduct(s3fp,a4fp);;
gap> prodrels:=RelatorsOfFpGroup(prod);;
gap> rels0010:=[prod.1*prod.3*prod.1^-1*prod.3^-1];
[ f1*f3*f1^-1*f3^-1 ]
gap> rels0020:=[prod.1*prod.4*prod.1^-1*prod.4^-1];
[ f1*f4*f1^-1*f4^-1 ]
gap> rels0030:=[prod.2*prod.4*prod.2^-1*prod.4^-1];
[ f2*f4*f2^-1*f4^-1 ]
gap> rels0040:=[prod.2*prod.3*prod.2^-1*prod.3^-1];
[ f2*f3*f2^-1*f3^-1 ]
gap> relscon:=Concatenation(prodrels,rels0010,rels0020,rels0030,rels0040);;
gap> relsstr:=String(relscon);
"[ f1^2, f2^2, f1*f2*f1*f2*f1*f2, f3^3*f4^-3, f3^3*f4^-1*f3^-1*f4^-1*f3^-1, f3\
^-1*f4^-1*f3*f4*f3^-1*f4^-1*f3*f4, f1*f3*f1^-1*f3^-1, f1*f4*f1^-1*f4^-1, f2*f4\
*f2^-1*f4^-1, f2*f3*f2^-1*f3^-1 ]"
gap> f/EvalString(relsstr);
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> dirprodfp:=last;
<fp group on the generators [ f1, f2, f3, f4 ]>
gap> dirprod:=DirectProduct(s3,a4);
Group([ (1,2,3), (1,2), (4,5,6), (5,6,7) ])
gap> IsomorphismGroups(dirprodfp,dirprod);
[ f1, f2, f3, f4 ] -> [ (1,2), (1,3), (4,7,6), (4,5,6) ]
gap> 

Thank you, Rudolf Zlabinger