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. 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? 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
