In Rob's example it would be nice (and good for the students to see)
if instead of looping over all g in G to construct conjugate subgroups
he could loop over coset representatives of H in G (where H is his
representative_subgroup), or (optimal) coset reps of the normalizer of
H in G. Now we can already compute HG=H.normalizer(G) in Sage
directly, but I cannot see a way of getting a list of coset reps.
Presumably Gap can do this, so perhaps wrapping that function would be
worthwhile. I did a search_src("transversal") and indeed found (in
the file coding/binary_code.pyx) reference to a Gap function
RightTransversal.
John Cremona
2008/9/24 Carlo Hamalainen <[EMAIL PROTECTED]>:
>
> On Wed, Sep 24, 2008 at 7:29 PM, Rob Beezer <[EMAIL PROTECTED]> wrote:
>>
>> A homework exercise for my students asks them to find all subgroups of
>> S_4, which should be a very instructive exercise, even if a bit
>> unreasonable. In SAGE, the conjugacy_classes_subgroups() method
>> gets you started, and the quick-and-dirty brute-force code below
>> creates all possible subgroups by doing the necessary conjugations.
>
> GAP has code for finding low index subgroups. Here's some Sage code to
> find all subgroups up to a certain index in the (3,3,3) triangle
> group:
>
> ##################################
> p = 3
> q = 3
> r = 3
>
> F = gap.new("FreeGroup(3)")
> gens = list(gap.GeneratorsOfGroup(F))
> rho1 = gens[0]
> rho2 = gens[1]
> rho3 = gens[2]
>
> triangle_group = F.FactorGroupFpGroupByRels([F.1*F.2*F.3, \
> F.1**p, F.2**q, F.3**r])
>
> iter = gap.LowIndexSubgroupsFpGroupIterator(triangle_group, \
> gap.TrivialSubgroup(triangle_group), 30)
>
> while str(gap.IsDoneIterator(iter)) == 'false':
> H = gap.NextIterator(iter)
>
> nr_points = gap.Index(triangle_group, H)
>
> G = gap.Image(gap.ActionHomomorphism(triangle_group, \
> gap.RightCosets(triangle_group, H), gap.OnRight))
>
> print gap.Order(G)
> ##################################
>
>
> I have plans to write a Cython wrapper for LOWX some time, see
> http://designtheory.org/~peter/software/lowx/
>
> --
> Carlo Hamalainen
> http://carlo-hamalainen.net
>
> >
>
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