[GAP Forum] strange result of FactorGroupFpGroupByRels

[Derek Holt]

Dear Hebert, Forum,

On Wed, Nov 18, 2015 at 04:25:30PM +0100, Hebert Pérez-Rosés wrote:
> Dear Forum,
> 
> I am working with the group 17 of order 108, from the small group library,
> which I henceforth denote by H. I am getting a strange result when I try to
> factor H, and I wonder if you could help me find an explanation.
> 
> The subgroup I, generated by the last generator, F.5, is a normal subgroup
> of order 3, and indeed, the factor group H/I is S_3 x S_3, of order 36.
> 
> Now, I my understanding is that the group J, generated by [F.1, F.2, F.3,
> F,4] should have index 3 in H, but GAP tells me this index is 1. I am
> including the code below for reference.

There is no reason to expect J to have index 3 in H.

> Ultimately, I presume that H is a semidirect product of S_3 x S_3 and C_3,
> and I would like to find the homomorphism  phi: S_3 x S_3 ----> Aut(C_3).
> How can I do that?

But it isn't a semidirect product - it is a non-split extension. So in
fact there was no possibility that the group J could have has index 3 in H.
You can verify that in GAP as follows:

gap> G:= SmallGroup(108,17);;
gap> H:=Subgroup(G,[G.5]);;
gap> ComplementClassesRepresentativesEA(G,H);
[  ]

There is however still a well defined homomorphism
phi: S_3 x S_3 ----> Aut(C_3)
defined by conjugation.

Regards,
Derek Holt.


> By the way, I am using GAP 4.4.12.
> 
> Best regards,
> 
> Hebert Pérez-Rosés,
> University of Lleida, Spain
> 
> ===================================
> 
> gap> G:= SmallGroup(108,17);
> <pc group of size 108 with 5 generators>
> 
> gap> H:= Image(IsomorphismFpGroup(G));
> <fp group of size 108 on the generators [ F1, F2, F3, F4, F5 ]>
> 
> gap> RelatorsOfFpGroup(H);
> 
> [ F1^2, F2^-1*F1^-1*F2*F1, F3^-1*F1^-1*F3*F1, F4^-1*F1^-1*F4*F1*F4^-1,
>   F5^-1*F1^-1*F5*F1*F5^-1, F2^2, F3^-1*F2^-1*F3*F2*F3^-1,
> F4^-1*F2^-1*F4*F2,
>   F5^-1*F2^-1*F5*F2*F5^-1, F3^3, F4^-1*F3^-1*F4*F3*F5^-1,
> F5^-1*F3^-1*F5*F3,
>   F4^3, F5^-1*F4^-1*F5*F4, F5^3 ]
> 
> gap> I:= FactorGroupFpGroupByRels(H,[H.5]);
> <fp group on the generators [ F1, F2, F3, F4, F5 ]>
> 
> gap> StructureDescription(I);
> "S3 x S3"
> 
> gap> J:= FactorGroupFpGroupByRels(H,[H.1,H.2,H.3,H.4]);
> <fp group on the generators [ F1, F2, F3, F4, F5 ]>
> 
> gap> StructureDescription(J);
> "1"
> 
> # At this point I thought that GAP's answer was due to the fact that the
> subgroup generated by [H.1, ..., H.4] was not normal, but when I tried to
> verify this conjecture, I got:
> 
> gap> S:= Subgroup(H, [H.1,H.2,H.3,H.4]);
> Group ([ F1, F2, F3, F4 ])
> 
> gap> IsNormal(H, S);
> true
> 
> gap> Index(H, S);
> 1
> 
> # Where is the problem here? Have I missed something?
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