Serg wrote:What "5 transformations above" do you mean? What do you mean saying "these transformations to exist in the first place"?
I meant the same 5 grid operations that you listed in your "computational approach" description above. It just happens that these can be defined with exactly 5 transformations (permutations), 1 for transposition, 2 for band permutations, and 2 for rows within 3 bands. That is, the SudokuP symmetry group G is completely generated by just 5 permutations:
- transposition
- swap bands 1 and 2
- swap bands 2 and 3
- swap rows (1,2) + (4,5) + (7,8)
- swap rows (2,3) + (5,6) + (8,9)
All required SudokuP-preserving grid permutations can be made with combinations of these.
By "exist" I just meant to say "defined by the allowable transformations". Just another way of saying that if isomorphism is based on these group permutations, then the orbit problem doesn't apply.
And that seems to be the key point, I just naturally assumed that if we define a group with specific set of SudokuP transformations, then we automatically define isomorphism as equivalence under those transformations, not the full set of Sudoku transformations.
I think we all agree that under this definition, the symmetry group is well-defined, orbits don't intersect, Burnside's Lemma applies, and all is right in the universe!
But if we define isomorphism as equivalent under Sudoku transformations, we definitely have orbits that intersect, and our universe is chaotic (and, alas, no Burnside!).
But the ordered universe, you say, is less interesting! In any case we clearly need to distinguish between these 2 isomorphism definitions, and I suggest we use p-isomorphism for the restricted (but uninteresting) case, and where necessary s-isomorphism for standard Sudoku isomorphism. "Isomorphism" unqualified will be assumed to be "s-isomorphism".
So the debate really boils down to this question:
- Should we define "essentially different" SudokuP grids in terms of p-isomorphism or not? (discuss)
