Martin Baker wrote:
>
> On 01/25/2017 05:23 AM, Waldek Hebisch wrote:
> > I also noticed that Todd-Coxeter (toPermutationIfCan)
> > may produce something strange. Namely, the following
> > (junky) relations:
>
> Do you think there is a need for a simplifier for PermutationGroup like
> this:
> http://dracos.co.uk/maths/permutations/
> Or does representation in terms of strong generators make this
> unnecessary/difficult?
I do not see what the page at your link is supposed to do:
there is a box to enter a permutation, but what we deal
with is interactions between several permutations.
> Either way I guess it would be more efficient and scalable if some
> simplification could also be done inside the Todd-Coxeter algorithm. I
> know that the Todd-Coxeter code does not yet detect and remove
> 'coincidences', that is, duplicated points. So this in the next
> improvement that needs to be made.
I do not understand what you wrote above: the whole point
of Todd-Coxeter is to recognise when words represent the
same coset. If your routine is unable to correctly
find out if cosets are equal, then it is very serious
problem (basicaly it would mean that the routine does not
work).
ATM I do not know what is good strategy when implementing
Todd-Coxeter. One thing that I know is that it is
useful to work with cosets with respect to a nontrivial
subgroup.
Concerning strong generators, in principle given presentation
in terms of strong generators it is easy to rebuild the
stabilizer chain and obtain permutation representation
quite close to orignal one. I am not sure if Todd-Coxeter
should do this automatically, but at least should allow
specifying subgroups by hand.
Some more remarks about presentatins: doing calculations
by hand I frequently depend on specific special relations,
like order of elements or comutation relations. Or on
lack of relations: if there is no relations involving given
generator, then the group is free product of Z with
group in other generators and relations. If generator
commute with other generators, but there are no other
relations, then we have usual product. Applying such
reasoning to a subgroup generated by subset of generators
(or more generally by some subwords) can give a subgroup
with know structure which may help solving whole problem.
I am not sure if programs should try do do such reasonings
automatically, because they are only applicable in special
situations.
BTW: We probably should collect several examples of
presentatins and ways to handle them. Ideally FriCAS
should be able to handle (possibly using a database
of standard tricks) various popular presentations.
--
Waldek Hebisch
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