[GAP Forum] Presentation of a group

[Derek Holt]

Dear William, Forum,

I think possibly you are being a little naive in expecting GAP to be able to
compute the order of these groups just starting from a presentation. GAP
will attempt to find the order using coset enumeration, either over the
trivial subgroup or over a cyclic subgroup. These two groups have orders
244823040 and 4030387200. The smallest indexes of cyclic subgroups are
(if I have got this right) 10644480 and 143942400, but there is no
guarantee that GAP would find a cyclic group of smallest index.

While these numbers are not completely outside the range of coset enumeration,
you would need a lot of memory to carry them out. So the messages you are
getting from GAP are not surprising - I have not checked your presentation, but
there is no reason to believe that there is mistake in it.

You would do better to look for subgroups of smaller index, possibly
defined on a subset of the generators, and attempt coset enumeration
directly over those subgroups, rather than just asking GAP for the order.
But you would need to take into account that if you did that and it
worked, and the image of the permutation action calculated has the
expected order, then you would still need to show that action was core-free.
You could attmept that either by using the visible relations that involved
only those subgroup generators, or by computing a presentation of the
subgroup and then calculating its order.

Best wishes,
Derek Holt

On Thu, Nov 30, 2017 at 02:37:34PM +0000, William Giuliano wrote:
> Dear Forum,
>                     I have written the presentation of two groups, which I
> am quite sure are the Mathieu group M_24 and the Held group. The problem I
> have is the message I get when I try to find their orders:
> 
> #I  Coset table calculation failed -- trying with bigger table limit
> 
> #I  Coset table calculation failed -- trying with bigger table limit
> 
> Error, reached the pre-set memory limit
> 
> (change it with the -o command line option) in
> 
>   g[2 * limit] := 0;
> 
> 
> I don't know if this is an indication of an actual error in my presentation
> or something which has to do with GAP. Here is one of the presentations
> 
> gap>
> f:=FreeGroup("a_1","a_2","a_3","a_4","a_5","a_6","a_7","a_8","a_9","a_10","a_11","a_12","a_13");
> 
> <free group on the generators [ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8,
> a_9, a_10, a_11, a_12, a_13 ]>
> 
> gap>
> r:=[f.1^2,Comm(f.1,f.3)*f.2^(-1),Comm(f.1,f.5)*f.4^(-1),Comm(f.1,f.8)*f.7^(-1),Comm(f.1,f.9),Comm(f.1,f.10),Comm(f.1,f.12),Comm(f.1,f.13)*f.6^(-1),
> 
> >
> f.2^2,Comm(f.2,f.6)*f.4^(-1),Comm(f.2,f.8),Comm(f.2,f.9)*f.7^(-1),Comm(f.2,f.10),Comm(f.2,f.11)*f.1^(-1),Comm(f.2,f.13)*(f.4*f.5)^(-1),
> 
> >
> f.3^2,Comm(f.3,f.6)*f.5^(-1),Comm(f.3,f.7),Comm(f.3,f.9)*f.8^(-1),Comm(f.3,f.10),(f.3*f.11)^3,Comm(f.3,f.13)*f.4^(-1),
> 
> >
> f.4^2,Comm(f.4,f.8),Comm(f.4,f.9),Comm(f.4,f.10)*f.7^(-1),Comm(f.4,f.12)*f.2^(-1),
> 
> >
> f.5^2,Comm(f.5,f.7),Comm(f.5,f.9),Comm(f.5,f.10)*f.8^(-1),Comm(f.5,f.11)*f.6^(-1),Comm(f.5,f.12)*f.3^(-1),
> 
> > f.6^2,Comm(f.6,f.7),Comm(f.6,f.8),Comm(f.6,f.10)*f.9^(-1),(f.6*f.12)^3,
> 
> > f.7^2,Comm(f.7,f.11),Comm(f.7,f.12),Comm(f.7,f.13)*f.4^(-1),
> 
> > f.8^2,Comm(f.8,f.11)*f.9^(-1),Comm(f.8,f.12),Comm(f.8,f.13)*f.5^(-1),
> 
> > f.9^2,Comm(f.9,f.12)*f.10^(-1),Comm(f.9,f.13)*f.6^(-1),
> 
> > f.10^2,Comm(f.10,f.11),(f.10*f.13)^3,
> 
> > f.11^2,(f.11*f.12)^4,Comm(f.11,f.13)*(f.6*f.4)^(-1),
> 
> > f.12^2,(f.12*f.13)^5,
> 
> > f.13^2,(f.6*f.12*f.13)^5,(f.10*f.13*f.12)^5];;
> 
> gap> g:=f / r;
> 
> <fp group on the generators [ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9,
> a_10, a_11, a_12, a_13 ]>
> 
> 
> Thanks
> Best regards,
> William
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