The New Sudoku Players' Forum

[Sojourner9]

Hi All,

Because of the the work I was doing regarding reorderings (3359232) I decided to go back and look at generating conjugacy classes using GAP.

Fraser's Home Page has moved. I updated the URL on Wikipedia. There he shows the input file and commands he typed into GAP to get the 275 conjugacy classes.
His input file, as described in his paper used a 9x9 array with the cells labeled 1 thru 81.
His input file shows cyclic notations of how he moved the cells around to described the reordering of the sudoku grid.
He never actually tell GAP that he is presenting a 9x9 array, just the moves.

As many may have done already, you can look at this input and figure out what he was trying to do.
And when completed you can look at the output of GAP to see what each class is doing.
The orderings he started with are:

1. Transposition on the upper left to lower right axis.
2. 90 degree rotation.
3. swap columns 1 and 2.
4. swap columns 2 and 3.
5. swap columns 4 and 5.
6. swap columns 5 and 6.
7. swap columns 7 and 8.
8. swap columns 8 and 9.
9. swap stacks 1 and 2.
10. swap stacks 2 and 3.

He didn't include anything for rows and bands because transposition took care of that, I assume.
I don't remember when I learned that I don't need a "swap columns 1 and 3" as repeated use of 3 and 4 gives all six permutation.
Of this Item 2 is not needed as the same can be accomplished with transposition and reflection of the columns.
I have thought for a long time that items 5, 6, 7, and 8 were also not needed.

But now to what I have done.

I repeat the exercise only this time I have rows and columns, like this:

Code: Select all

             Columns
        1 2 3 4 5 6 7 8 9
     10
     11
     12
     13
Rows 14
     15
     16
     17
     18

As with the original, we never actually tell GAP what we are representing, just how it moves.

So now I can provide the following input to GAP:

Code: Select all

sudoku_18 := Group( (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18), (1,2), (2,3), (1,4)(2,5)(3,6), (4,7)(5,8)(6,9) );


This has all the "redundent" orderings removed.
Running this I still get 275 conjugacy classes and their counts.
The only problem is that the order of the conjugacy classes changes.
I tried adding back (4,5), (5,6), (7,8), and (8 9) and I could not get the order corrected I even added the 90 degree turn and could not get it. I can still try later.

So a 90 degree turn should be transposition composed with reflection of the columns which should be:
(1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) ∘ (1,9)(2,8)(3,7)(4,6)

Working right to left, 1>9>18, 2>8>17, 3>7>16, 4>6>15, 5>14, 6>4>13, 7>3>12, 8>2>11, 9>1>10.
So the 90 degree rotation reordering should be:

(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)

Maybe I need to reverse the composition. But anyway here is an example of the conjugacy classes:

Code: Select all

  216     (10,12,11)(13,17,14,16)(15,18)
  648     (1,2,3)(13,17,14,16)(15,18)
  2916    (2,3)(8,9)(13,16)(14,18,15,17)
  5832    (1,2)(8,9)(10,11,12)(13,16)(14,18,15,17)
  11664   (2,3)(4,5,6)(8,9)(10,11,12)(13,16)(14,18,15,17)
  5832    (1,3)(4,5,6)(8,9)(13,16)(14,18,15,17)
  108     (11,12)(13,16)(14,17)(15,18)
  648     (1,2,3)(10,11)(13,16)(14,17)(15,18)
  1296    (1,2,3)(4,5,6)(10,12)(13,16)(14,17)(15,18)
  864     (1,2,3)(4,5,6)(7,8,9)(11,12)(13,16)(14,17)(15,18)
  216     (10,12)(13,17,14,18,15,16)
  1296    (1,2,3)(11,12)(13,17,14,18,15,16)
  2592    (1,2,3)(4,5,6)(10,11)(13,17,14,18,15,16)
  1728    (1,2,3)(4,5,6)(7,8,9)(10,12)(13,17,14,18,15,16)
  2916    (2,3)(8,9)(11,12)(13,16)(14,17)(15,18)
  5832    (2,3)(4,5,6)(8,9)(10,11)(13,16)(14,17)(15,18)
  5832    (2,3)(8,9)(10,12)(13,17,14,18,15,16)
  11664   (2,3)(4,5,6)(8,9)(11,12)(13,17,14,18,15,16)


The classes have no inherent ordering we would have to agree on a fixed input and then reorder our analyzes.
Unless I can fix the reordering problem.
I will try to show these in rowMap and colMap formats at some point.
Can we agree this is a much easier way to express the classes?

As can be seen, it is much clearer what is being moved in each conjugacy class, Agreed?

[blue]

So a 90 degree turn should be transposition composed with reflection of the columns which should be:
(1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) ∘ (1,9)(2,8)(3,7)(4,6)

Working right to left, 1>9>18, 2>8>17, 3>7>16, 4>6>15, 5>14, 6>4>13, 7>3>12, 8>2>11, 9>1>10.
So the 90 degree rotation reordering should be:

(1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)

The 90 degree turn should include 4-cycles.
Example: row1 -> col9 -> row9 -> col1 -> (loop)

Code: Select all

gap> q := (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) * (1,9)(2,8)(3,7)(4,6);
(1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14)
gap>

Added trivia: You can get by with three generators ... swap cols 1 & 2, swap stacks 1 & 2, and 90 degree rotation.

Code: Select all

gap> sg := Group( (1,2), (1,4)(2,5)(3,6), (1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14) );
Group([ (1,2), (1,4)(2,5)(3,6), (1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14) ])
gap> [ Size(sg), NrConjugacyClasses(sg) ];
[ 3359232, 275 ]


[Sojourner9]

The 90 degree turn should include 4-cycles.
Example: row1 -> col9 -> row9 -> col1 -> (loop)

Code: Select all

gap> q := (1,10)(2,11)(3,12)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18) * (1,9)(2,8)(3,7)(4,6);
(1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14)
gap>

Thanks Blue,
I added your 90 degree rotation into the code and added the redundants to emulate Frazer's input and I now see the counts in the expected order.
Now it is just a matter of verifying that they do the same work.

Added trivia: You can get by with three generators ... swap cols 1 & 2, swap stacks 1 & 2, and 90 degree rotation.

Code: Select all

gap> sg := Group( (1,2), (1,4)(2,5)(3,6), (1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14) );
Group([ (1,2), (1,4)(2,5)(3,6), (1,10,9,18)(2,11,8,17)(3,12,7,16)(4,13,6,15)(5,14) ])
gap> [ Size(sg), NrConjugacyClasses(sg) ];
[ 3359232, 275 ]


I'll bet the order is is not the same.
I wonder how (2 3) is accomplished?

It figures the first time I try to demonstrate how to do something I would get it wrong.
And now I can see how to get GAP to do the "multiplication" for me.
I haven't done composition where all the numbers aren't in both entries. Is composition not the correct thing to do here?
Math isn't my first language and I only know about groups what I read in books.
I tried to audit a class is abstract algebra last fall and they directed me to concrete math instead. I will try again this fall.
I was trying to do composition to combine two permutations but obviously that is not what I should have done, or did I do it wrong.
It makes sense that there should be 4-cycles since it is a rotation, which is what I like about group theory. It is common sense and alchemy that works.

[Sojourner9]

I think the ordering of the 275 is still not right, the 90 degree entry is at 117 instead of 86.

[blue]

I haven't done composition where all the numbers aren't in both entries. Is composition not the correct thing to do here?
(...)
I was trying to do composition to combine two permutations but obviously that is not what I should have done, or did I do it wrong.

"Composition" is right. You did it wrong, I guess.
Here's how one of the 4-cycles comes about.
The LHS arrow is from the "transpose" operation, and the RHS is from the horizontal reflection.

Code: Select all

 1 -> 10 -> 10
10 ->  1 ->  9
 9 -> 18 -> 18
18 ->  9 ->  1      (1,10,9,18)

BTW: I consider myself an amateur mathematician, at best.
Don't rely on me for correct terminology, etc.
I have to figure out GAP again, every time I (re)install it.
My group theory knowledge comes from Wikipedia and a 40 years ago, 9-week course, where the textbook was from the "Schaum's Outline" series.

[Sojourner9]

What is returned here is not immediately useful. If we don't have transposition in the class then 10 thru 18 becomes rowMap 1 thru 9 and 1 thru 9 remains colMap 1 thru 9.
When I deal with transposition in my code I just swap the rows and column indexes when I access the array.
So it makes no sense to have a cyclic notation where we swap one column with one row, except it swaps them all at the same time.
But I'll have to sort this out.

The next question is, Why do I want to do this?
I guess I thought that having a simpified 275 would help identify the classes in the reorderings of the grid, like coloin was talking about.
Also I wanted to show that I could convert the 275 to my rowMap/colMap since it is just one permutation of the 3359232 reorder group.

The thing is the conjugacy classes are not groups, just a single motion. You could consider these as generators and build up subgroups by repeated application.
So I will stop here because I have reached the limits of my understanding.

[Sojourner9]

Hi all,

I was not able to solve the problem I was having with GAP.
No matter what input I gave it using the 18 element input I could not get an analogous output to that of the 81 element input.
The count for a class indicates there are that may different ways of representing the same thing.
I still got 275 classes and the same counts for each but the ordering of the classes and the representatives changed.
Even the 81 element classes will change order and representatives if the input changes.
So the order everyone is used to for the 275 is completely dependent on running that specific input.

Not sure why I can't get it to work. It is like a hash table/associative arrays/dictionaries in programming languages where they say not to expect the keys to be in a certain order.
I tried to take the 275 output for an 18 element input and compare the cyclic notations of each but that is when I found that the representatives for each had changed.

I took the original 275 output and converted that to rowMap/colMap format.
This was pretty easy I made a 9x9 array and filled with 1...81 and then did the action described by the class representative.
The I just looked for the row and column in the array that contained the number 1 and just read off the row which was 1...9 and the column which was 1,10,19,28,37,46,55,64,73 and converted that to 10...18.
If the row and column were swapped I changed the sign of the colMap.

Because it might be of use to someone, here is the Original 275 Conjugacy Classes in rowMap/colMap format.

Hidden Text: Show

Code: Select all

Class   1. 123456789  123456789 Size      1, 18383222420692992
Class   2. 123456978  123456978 Size     36, 0
Class   3. 123456897  123456789 Size     12, 0
Class   4. 123456978  123645978 Size    144, 0
Class   5. 123456789  123645978 Size     24, 0
Class   6. 123645978  123645978 Size    144, 0
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Class   8. 123456789  312645897 Size     16, 107495424
Class   9. 123645897  312645978 Size    192, 4204224
Class  10. 312645978  312645978 Size     64, 2508084
Class  11. 123465798  123465798 Size    729, 0
Class  12. 123465879  312465798 Size   2916, 0
Class  13. 312465879  312465879 Size   2916, 0
Class  14. 123456789  132465789 Size     54, 0
Class  15. 123456978  132465978 Size    648, 0
Class  16. 123456897  132465789 Size    324, 0
Class  17. 123456789  132465978 Size    108, 0
Class  18. 123645978  132465978 Size   1296, 0
Class  19. 123645897  132465789 Size    648, 0
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Class  27. 456789231  123456789 Size    288, 5184
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Class  29. 456789312  123645978 Size   3456, 1296
Class  30. 456789123  123645978 Size   1728, 294912
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Class  45. 123456798  123456798 Size     81, 0
Class  46. 123456798  123645798 Size    648, 0
Class  47. 123645798  123645798 Size   1296, 0
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Class  56. 456798123  123456798 Size   3888, 0
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Class 167. 132465987  312645978 Size    432, 0
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For some random reason I was looking at Mathmagics eulogy thread and it took me to the reddit equivalent.
Anyway this took me to his posts on math.stackexchange.
Looking as some of his posts I found this one List Conjugacy Classes in GAP?
Here is is asking the question about how to see all the representatives for a class.

As a last ditch effort, I tried to take the original 275 and make a 18 element cyclic notation equivalent starting with the rowMap/colMap list from above.
So I have to figure out how to do the composition correctly. and all in all it was a real pain to write and it took me a week.
I learned that the 18 element is like a camera looking at a never changing grid where the 81 element classes are a fixed camera and it is moving the grid.
So to class that represents a 90 degrees clockwise rotation of the grid should be represented by a class that represents a 90 degree counter-clockwise rotate using the 18 elements. Or is that anti-clockwise?
I fed this list into GAP and was going to try to check each entry against all the representative of all the classes to try to figure out the scramble table for the classes.
But I just got tired of working on this and did not want learn to program GAP that much.

I am on the precipice of a rant about GAP, but I will try to keep my cool.
To use GAP you have to enter the GAP environment where it give you a prompt and you have to type in your entire code.
Even if you save off all the code outside of GAP and use cut and paste on the input, you still have to write the code and process the output.
Say asked for all the members of a class I still have to copy and paste the output into an editor to make the it pretty.
And then I have to learn the GAP programming language, GAPanese?
I already have a really good language that can maniputate the output and format the display.
The work of editing the output to make it compatible input to my program is ridiculous.

Has anyone been able make GAP as a library into your favorite programming language?
Or at least found a better way to work with GAP then what I described?

[BlazeRaven]

When you mention the order of the conjugacy classes changing depending on the input, does that affect how we interpret the classes themselves, or is it just a matter of reordering them for easier comparison?