Cosets changes #126
Cosets changes #126
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| Return the coset `gH`. | ||
| !!! note | ||
| Since GAP supports right cosets only, the underlying GAP object of `left_coset(H,g)` is the right coset `H^(g^-1) * g`. |
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While this describes what the code does, this strikes me as deeply problematic. It means that you have to creates lots of new subgroups by conjugating H around.
My idea would be to represent gH by Hg^{-1}, using that fact that the inversion map induces a bijection between them.
That said, I am actually still sceptical about the whole idea of providing left cosets. Perhaps it's a good idea to do so, but I'd really like to see some concrete applications in order to figure out what the "right" way (or at least a "somewhat useful way") to go about them might be.
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My idea would be to represent gH by Hg^{-1}, using that fact that the inversion map induces a bijection between them.
This would be much faster during the creation of the coset (I don't have to create the conjugate subgroup), but would it be slower every time I need to access to it?
If lc=gH, and I save it as Hg^-1, every time I type something like rand(lc) or elements(lc), or some intersection involving lc, I need to compute an inverse. A possible syntax for the rand function could be
> function rand(lc::Coset)
> x=rand(lc)
> if isleft(lc) then x=x^-1 end
> return x
> end
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I think it is perfectly realizable, maybe with a little care about the fact that the underlying GAP object is not the object I'm interested (but its image under the inverse function), but is it really faster?
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Well, that exactly is my point: what is "faster" depends entirely on the use case. But no matter how you implement the left cosets, one thing stays the same: as long as they simply (ab)use GAP's right cosets, it will always be faster if you rewrite your code to use right cosets instead of left cosets.
Anyway: as long as this is an implementation detail is hidden from the user, this is fine. But I would not want to mention it to the user! The user should not have to care about this.
Next thing: GAP doesn't really do much with RightCosets anyway. So we could think about simply not using GAP objects of type RightCoset at all, and simply implement our own two types RightCoset and LeftCoset, which simply wrap a group element g plus a subgroup H. Implementing things like rand or elements then is easy: e.g. rand(H) * g to get a random element. I'll put this idea into another issue
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It seems good. Do I try this new type on the same branch? Or do we open a new one?
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I want to merge the branch
coset-changesinto the master branch before going on with the work in the manual. This branch deals the questions raised in the Issue #118. In particular:isrightcosetandas_right_cosetthat says whether a coset is right and turning a coset into a right coset. Same for the left.x * yH --> (xy) * Hand similar ). Now it is also possible to define a coset just multiplying an element by a coset (e.g.x * H --> left_coset(H,x)).