On Thu, Apr 11, 2013 at 01:09:59AM +0200, Borie Nicolas wrote:
> >I wonder if there is a way to get a canonical form of a subgroup of a
> >permutation group (or, even better, any group). This would be
> >something like a method "canonical_labeling" for permutation groups
> >that returns an isomorphic permutation group, and such that two groups
> >are isomorphic if and only if their "canonical labellings" coincide.
> If I would have to do it myself, I will use the following process :
> a strong generating system (s.g.s.) identify uniquely a permutation
> group. building a s.g.s. is a polynomial algorithm and allow after
> to have a lot of nice features for close to free (cardinality,
> contains tests, ...). Unfortunately, a s.g.s is not unique but
> exploiting the fact that the lexicographic order is a total order
> over permutations, you can define a minimal s.g.s for a group. Now,
> two permutation groups (define by whatever : generators, list of
> elements, ...) would be isomorphic if and only if they have the same
> minimal s.g.s. (for information, the symmetric group of degree n S_n
> have s.g.s. of size maximum among all subgroup of S_n and you need
> to keep in memory binomial(n,2) permutations of size n).
If I understood Christian's question right, he does not only want to
test if the two groups are equals as sets of permutations, but if they
are isomorphic as groups. So for him the following groups generated by
permutations would all be "the same":
- < (1,2) >
- < (2,3) >
- < (1,2)(3,4) >
- the multiplicative group ({-1,1}, *).
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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