[GAP Forum] strange result of FactorGroupFpGroupByRels

[Hebert Pérez-Rosés]

Thanks a lot, Derek. That explains everything.



2015-11-18 16:58 GMT+01:00 Derek Holt <D.F.Holt at warwick.ac.uk>:

> 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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