Group cosets: implement custom in, improve iterator

[fingolfin]

Right now in for group cosets resorts to iterating over the coset, which is very inefficient.

We should provide a custom method for this. I would not dispatch to GAP's, but instead implement this "directly": i.e. to test g in Hx, test if g/x in H and similar for g in xH (so that we avoid creating GapObj for those).

---

Speaking of iterators, the iterate implementation for GroupCoset currently directly wraps a GAP coset iterator. But (a) this requires creating a GapObj (with extra overhead for left cosets). And furthermore on the GAP side, the iterator is a wrapper for an enumerator.

Instead, the Julia iterator for a coset C could be based on the group iterator for C.H.

---

Lastly, contrast

Base.IteratorSize(::Type{<:GAPGroup}) = Base.SizeUnknown()
Base.IteratorSize(::Type{PermGroup}) = Base.HasLength()

versus

Base.IteratorSize(::Type{<:GroupCoset}) = Base.SizeUnknown()

I suggest we use something like this:

Base.IteratorSize(::Type{<:GroupCoset{T}}) where T = Base.IteratorSize(T)

And then I assume similar for GroupDoubleCoset.

[ThomasBreuer]

For a coset Hg or gH of type GroupCoset{T, S}, T is not necessarily the type of H.
If we want to set Base.IteratorSize for cosets depending on Base.IteratorSize for the type of H then I think we have to change the parameterization to GroupCoset{TG, TH, S} where TG and TH are the types of G and H, respectively.

Concerning the iterator for double cosets, one solution would be to use the functionality of GAP's RepresentativesContainedRightCosets (which is also used by GAP's membership test for double cosets), and to delegate to the iterator for right cosets.

[ThomasBreuer]

Where are we now:

With #4302, we get

• in for cosets delegating to in for groups, and in for double cosets delegating to GAP
• an extended parameterization of the GroupCoset type such that the idea to get a better Base.IteratorSize for cosets works

Concerning double cosets, we would need more GAP functionality on the Oscar side in order to replace the delegations to GAP by something better. For both iteration and membership tests for double cosets, my idea would be to split the double coset into right cosets and to delegate to these object, as is done in GAP. For that, GAP's RepresentativesContainedRightCosets could be exported to Oscar. (I am not sure whether GAP's approach using canonical coset representatives is generically a good idea, it defaults to unpacking right cosets and taking the minimal element.)