Add a constructive membership test for double cosets

[ThomasBreuer]

• introduce a new type field right_coset_reps for double cosets, which gets computed on demand, using the same ideas as GAP.Globals.RepresentativesContainedRightCosets but additionally storing the transforming elements
• introduce decompose for writing a given element in U*g*V in the form u*g*v, with u in U and v in V
• use the new data also in the nonconstructive membership test with in
• extend the tests accordingly

(The code still uses GAP.Globals.CanonicalRightCosetElement, which calls different methods for various kinds of group elements.)

[lgoettgens]

Do we already have a function like your new decompose for right cosets? If yes, it would make sense to use the same name and interface here.

[ThomasBreuer]

Do we already have a function like your new decompose for right cosets?

For right cosets, it is much simpler: If I have an element x and a right coset U*g then either x/g is the element in U we are looking for, or x is not in the coset.

[simonbrandhorst]

Thanks a lot. It looks good from my side.
An alternative name to decompose would be something like
is_element_with_data or is_in_with_data, constructive_membership_test. But that would set a general pattern.
So until we have settled once and for good on a name, I would suggest to call the function _decompose for the moment? Then we do not commit on a name yet (decompose is exported) and I can start using it for my purposes.

[fingolfin]

I am not sure if Julia can correctly deduce that it has to fetch the value in C.right_coset_reps[] only once afterwards and then not again(in lines 922, 923).

All in all, I'd suggest to use a pattern like this:

Suggested change

	if !isassigned(C.right_coset_reps)
	C.right_coset_reps[] = _right_coset_reps(C)
	end
	rcr = _right_coset_reps(C)

and then rename the current _right_coset_reps to, say, compute_right_coset_reps; and the new _right_coset_reps basically does

function _right_coset_reps(C)
  if !isassigned(C.right_coset_reps)
    C.right_coset_reps[] = _right_coset_reps(C)
  end
  return C.right_coset_reps[]
end