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

[Mike Newman]

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.
>
> --
> met vriendelijke groet / kind regards,
> nilo de roock
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