> < ^ From:

< ^ Subject:

Dear Mr Alvarez,

allow me first to make some general remarks about your computation

before I explain what happened in your GAP session.

I tried to compute a presentation of the normal closure of

the subgroup generated by the commutator of the two generators of a free

group on two letters with gap, and it appears to be enumerating something

which is infinite.

Let F be the free group on a and b. The normal closure of [a,b]

is the commutator subgroup F' of F and F/F' is a free abelian

group of rank two. This is an infinite group, of course.

Furthermore, F' is a free group of infinite rank. Therefore, any

presentation for F' has infinitely many generators.

The following is an outline of the method used in GAP for

computing a presentation for a subgroup H of a finitely presented

group G. First GAP attempts to compute a coset table for the

cosets of H in G via the modified Todd-Coxeter method. This can

only succeed if H has finite index in G. Once the coset table is

completed, a generating set and a set of defining relations is

constructed for H. In your situation, the computation of the

coset table is not possible, since the index of F' in F is

infinite.

Even if one could tell GAP somehow that the normal closure of

[a,b] is a free subgroup of F of infinite rank, there is nothing

interesting one could do with this subgroup in GAP because there

is practically no support for groups given by an infinite

generating set.

gap> F := FreeGroup("a",' "b") ;;

gap> E := F / [] ;;

gap> a := E.1 ;; b := E.2 ;;

gap> R := Subgroup( E, [a*b*a^-1*b^-1] ) ;;

gap> PresentationNormalClosure( E, R ) ;

Error, the coset enumeration has defined more than 64000 cosets:

type 'return;' if you want to continue with a new limit of 128000 cosets,

type 'quit;' if you want to quit the coset enumeration,

type 'maxlimit := 0; return;' in order to continue without a limit,

in

CosetTableFpGroup( F, TrivialSubgroup( F ) ) called from

PresentationNormalClosure( E, R ) called from

main loop

brk> return ;

At this stage, the Todd-Coxeter method has computed 64 000 cosets

an asks for confirmation to compute more. If you modify your

example slightly by adding the elements a^2 and b^2 as generators

to R, you will get a subgroup whose normal closure has finite

index (although R itself doesn't).

With kind regards,

Werner Nickel.

------------------------------------------------------------------------------ Dr (AUS) Werner Nickel Mathematics with Computer Science TU Darmstadt Tel: +49 (0)6151 163486 Fachbereich Mathematik, AG 2 Fax: +49 (0)6151 166535 Schlossgartenstr. 7 nickel@mathematik.tu-darmstadt.de D-64289 Darmstadt ------------------------------------------------------------------------------

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