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

Nilo de Roock ndroock1 at gmail.com
Tue Aug 22 10:04:01 BST 2006

```Dear Mike (Newman),

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.

--
met vriendelijke groet / kind regards,
nilo de roock
```