[GAP Forum] Unexpected Behaviour in Size()

[Alexander Hulpke]

Dear GAP-Forum,

Nilo de Roock wrote:

On Feb 26, 2006, at 9:39 AM, Nilo de Roock wrote:

> Hello GAP forum,
>
> I created the free group ( i got this spec. for the group from a  
> textbook-exercise ):
> gap> F:=FreeGroup(3);
> <free group on the generators [ f1, f2, f3 ]>
> gap> x:=F.1;
> f1
> gap> y:=F.2;
> f2
> gap> z:=F.3;
> f3
> gap> G:=F/[y^3*z^15,x^4*y^7*z^3,x^8,y^14,z^18];
> <fp group on the generators [ f1, f2, f3 ]>
>
> Then when I wanted to now the size of the group, GAP became a bit  
> of erratic.

I would not call this erratic.

This is the expected behaviour, as there are fundamental difficulties  
on algorithmic methods for finitely presented groups. (The so-called  
``word problem''.)
You might want to read the AMS notices article by 'Akos Seress  
(notices.ps on the page http://www.math.ohio-state.edu/~akos/ )
or the recent ``Handbook of Computational Group Theory'' by Holt et.  
al. for methods used and some of the fundamental problems arising.

GAP issues warnings that it is performing a lot (probably more than  
you expected) work, and still does not have a result. This is an  
indication that you might needs lots of memory or would be on the way  
of overloading your computer without ever getting a result -- GAP is  
trying to stop you doing something you don't really want.

In your case, GAP tries to compute the size of an fp group by  
calculating the index of a cyclic subgroup and rewriting the  
presentation. However an easy calculation (abelian invariants of G')  
shows that your group is infinite and cannot have any cyclic subgroup  
of finite index.

Best wishes,

     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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