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? > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
