[GAP Forum] AllGroups(273)[3] revisited.

[Dima Pasechnik]

there are typos in relations here: a^3=1 must be c^3=1 everywhere.

On Tue, 2006-08-22 at 11:04 +0200, Nilo de Roock wrote:
> Dear Mike (Newman),
> 
> Thank you very much for your reply!
> 
> 
> You wrote that
> 
> Put H1 = {a,b,c | a^7 = b^13 = a^3 = 1, ab = ba, a^c = a^2, b^c = b^3}.,
> 
> Put H2 = {a,b,c | a^7 = b^13 = a^3 = 1, ab = ba, a^c = a^2, b^c = b^9}.
> 
> are presentations for the groups of type C91 : C3 and are isomorphic to
> AllGroups(273)[3] and AllGroups(273)[4]
> 
> I tried to create these groups to have GAP confirm an isomorphism but until
> now I failed.
> 
> 
> For example, for A4, I would do it as follows. ( A4. a^3=1,b^2=1,
> a*b*a=b*a^-1*b. )
> 
> F:=FreeGroup(2);; a:=F.1;; gap> b:=F.2;;
> 
> H1:=F/[a^3,b^2,a*b*a*b^-1*a*b^-1];;
> 
> gap> Size(H1);
> 12
> gap> StructureDescription(H1);
> "A4"
> 
> gap> List(AllGroups(12),StructureDescription);
> [ "C3 : C4", "C12", "A4", "D12", "C6 x C2" ]
> gap> IsomorphismGroups(H1,AllGroups(12)[3]);
> [ f1, f2 ] -> [ f1*f3, f2*f3 ]
> 
> And then I get the isomorphism confirmed.
> 
> 
> 
> Now for H1 = {a,b,c | a^7 = b^13 = a^3 = 1, ab = ba, a^c = a^2, b^c = b^3},
> I tried the same.
> 
> gap> H1:=F/[a^7,b^13,a^3,a*b*a^-1*b^-1,a^c*a^-2,b^c*b^-3];
> <fp group on the generators [ f1, f2, f3 ]>
> gap> Size(H1);
> infinity
> gap>
> 
> I get a group of inifite order. What am I doing wrong here??
> 
> Any hints? Thanks on beforehand for any help.
>