[GAP Forum] Subgroups of Finitely Presented Groups

[Alexander Hulpke]

Dear Forum,

On Mar 10, 2011, at 11:43 PM, Jay Taylor wrote:
> 
> I have the following problem. I start with a free group F on a finite number
> of generators and then construct a finite finitely presented group H = F/R,
> for some relations R. I now define a subgroup S of H by giving a certain
> list of generators for S coming from H. When I call Elements(S) the elements
> are presented as words in the generators used to define S. However I would
> like to get GAP to express the elements of S as minimal length words in the
> original generators of F using the relations R. In other words I want to
> work with the image of S under an embedding of S into H. Is this possible?

This is possible if the presentation for H is well-behaved (in the sense that a knuth-bendix completion will terminate). This is for example the case if H is small. Before defining eny products in H, call
SetReducedMultiplication(H)
This will force all elements of H to be represented in a minimal form wrt. a ShortLex ordering and thus represent the elements of S as a minimal length word in the generators of H. For example

gap> F:=FreeGroup("a","b","c");
<free group on the generators [ a, b, c ]>
gap> AssignGeneratorVariables(F);
#I  Assigned the global variables [ a, b, c ]
gap> rels:=[a^2,b^2,c^2,(a*b)^3];
[ a^2, b^2, c^2, a*b*a*b*a*b ]
gap> H:=F/rels;
<fp group on the generators [ a, b, c ]>
gap> SetReducedMultiplication(H); 
gap> S:=Subgroup(H,[H.1^H.2,H.2]);    
Group([ a^-1*b^-1*a^-1, b ])
gap> Elements(S);
[ <identity ...>, a^-1, b^-1, a^-1*b^-1, b^-1*a^-1, a^-1*b^-1*a^-1 ]

Best,

   Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
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