[GAP Forum] Question about the groups C91:C3 in AllGroups(273).

[Jim Heckman]

Right.  Perhaps it will help Nilo to see the non-isomorphism by
noting that in H1 every element of order 3 conjugates every element
of order dividing 91 to either its 16th or its 74th power, while in
H2 every element of order 3 conjugates every element of order
dividing 91 to either its 9th or its 81st power.

-- 
Jim Heckman

On 21-Aug-2006, Mike Newman <newman at maths.anu.edu.au>
wrote in message <Pine.GSO.4.58.0608220851510.12643 at yin>:

> Dear Nilo,
>
> These groups are known to be difficult to tell apart.
>
> Let me first describe them in conventional notation.
>
> 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}.
>
> You will see the first is identified as [ 273,3 ] and
> the second as [ 273,4 ].
>
> An easy way to see they are different is to see what happens under a
> supposed isomorphism. With such a map, the Sylow 7-subgroups must agree
> and also the Sylow 13-subgroups. After that it is impossible to
> complete to an isomorphism.
>
> This is part of the theory of groups with square-free order which goes
> back to Otto Hoelder in 1895. In particular if p,q,r are primes such that
> p > q > r with r dividing both p-1 and q-1 then there are r-1 different
> non-abelian extensions of the cyclic group with order pq by the cyclic
> group with order r.
>
> Hope this helps,
>
>
>       Mike (Newman)
>
> On Mon, 21 Aug 2006, Nilo de Roock wrote:
>
> > Dear GAP users,
> >
> > I am using GAP 4.4.7, on a PC with Windows XP.
> >
> > For my research I need to know the difference between AllGroups(273)[3]
> > and
> > AllGroups(273)[4] and also how I can construct them other than using
> > AllGroups. Sofar I can only construct one and I can't find any group
> > property which is different between the two.
> >
> > My first question is about constructing C91:C3. AllGroups(273) yields.
> >
> > ["C13 x (C7 : C3)",
> > "C7 x (C13 : C3)",
> > "C91 : C3",
> > "C91 : C3",
> > "C273" ]
> >
> > I have been able to construct one instance of C91:C3 through
> >
> > gap> G:=CyclicGroup(IsPermGroup,3);
> > Group([ (1,2,3) ])
> > gap> N:=CyclicGroup(91);
> > <pc group of size 91 with 2 generators>
> > gap> A:=AutomorphismGroup(N);
> > <group with 2 generators>
> > gap>
> > f:=GroupHomomorphismByImages(G,A,[Elements(G)[2]],[Elements(A)[39]]);
> > [ (1,2,3) ] -> [ Pcgs([ f1, f2 ]) -> [ f1^4*f2^10, f2^9 ] ]
> > gap> NG:=SemiDirectProduct(G,f,N);
> > gap> NG:=SemidirectProduct(G,f,N);
> > <pc group of size 273 with 3 generators>
> > gap> StructureDescription(NG);
> > "C91 : C3"
> >
> > How can I construct the other C91:C3 ? ( I have tried mappings to other
> > elements in A ( C12 x C6
> > ) but none of them eventually results to a different C91:C3 in the
> > SemidirectProduct.
> >
> >
> >
> > My second question is about the difference is between AllGroups(273)[3]
> > and
> > AllGroups(273)[4], they seem isomorphic to me. I have tested various
> > properties and I can't figure out the difference.
> >
> > About AllGroups(273)[3] and AllGroups(273)[4]. I name the groups G1 and
> > G2...
> > gap> G1:=AllGroups(273)[3];
> > <pc group of size 273 with 3 generators>
> > gap> G2:=AllGroups(273)[4];
> > <pc group of size 273 with 3 generators>
> >
> > If the subgroups of G1, G2 are the same and the orders of the
> > elements... then where is the difference? I can't find it in GAP.
> >
> > The following commands illustrate some significant similarities
> > between the groups.
> >
> > gap> StructureDescription(G1);
> > "C91 : C3"
> > gap> StructureDescription(G2);
> > "C91 : C3"
> > gap> List(ConjugacyClassesSubgroups(G1),Representative);
> > [ Group([ ]), C3, Group([ f2 ]), Group([ f3 ]), Group([ f1, f2 ]),
> > Group([ f3, f1 ]), C91, Group([ f3, f2, f1 ]) ]
> > gap>
> > List(List(ConjugacyClassesSubgroups(G1),Representative),StructureDescript$
> > gap> $cription);
> > [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
> > gap>
> > List(List(ConjugacyClassesSubgroups(G2),Representative),StructureDescript$
> > gap> $cription);
> > [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
> > gap> $cription);
> > [ "1", "C3", "C7", "C13", "C7 : C3", "C13 : C3", "C91", "C91 : C3" ]
> > gap> Sum(List(Elements(G1),Order));
> > 7297
> > gap> Sum(List(Elements(G2),Order));
> > 7297
> >
> > Thanks on beforehand for any hints on this particular issue.