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Distance preserving automorphisms

Submitted by Joe Miller on Mon, 07/18/2005 - 20:30.
I have several questions:

Have the number of distance preserving automorphisms of the Rubik's cube and/or Junior cube been counted/enmerated? Is there a way to count/enumerate them with GAP?

Thank you for your time,
-Joe Miller
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AutomorphismGroup of 2x2x2 cube

Submitted by cubex on Sun, 07/24/2005 - 12:28.
I used the AutomorphismGroup function in GAP on the 2x2x2 cube. GAP returns Size(cube) where cube is the 2x2x2 cube of 88179840. GAP considers whole cube orientations in space as distinct. At first I thought that the AutomorphismGroup calculation might be intractible but after a few minutes it finished. Size(AutomorphismGroup(cube)) returns a group of size 176359680 with 8 generators. Unfortunately I have no idea about how to calculate distance preserving automorphisms with GAP. I'll email Martin, maybe he knows.
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1994 Cube-Lovers Articles on Distance Respecting Automorphisms

Submitted by Jerry Bryan on Sun, 07/24/2005 - 21:44.
Check out http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Martin_Schoenert__Distance_Respecting_Automorphisms.html And in turn, check out http://www.math.rwth-aachen.de/~Martin.Schoenert/Cube-Lovers/Martin_Schoenert__Re__The_real_size_of_cube_space.html

If I'm understanding the two articles correctly, Martin does not say how to calculate distance respecting automorphisms in GAP, but he does confirm that M (the set of rotations and reflections of the cube) is the largest subgroup of the outer automorphism group that respects distances.
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2x2x2 vs. Corners of 3x3x3

Submitted by Jerry Bryan on Sun, 07/24/2005 - 21:30.
The group of order 88179840 is the corners of the 3x3x3, not the 2x2x2. The order of the 2x2x2 is 3674160, which is 24 times smaller than the order of the 3x3x3.

The most conventional way to model the 2x2x2 is to start with the corners of the 3x3x3 and to fix one of the corners. Any move on the 2x2x2 can be made either on a particular face or on the opposite face. For example, the move R is equivalent to the move L on the 2x2x2. So suppose that you choose to fix the flu cubie. To do that, when you could make move F or B, always choose B. When you could make move F' or B', always make B'. When you could make L or R, always make R. Etc.

The result in the quarter turn metric will be six generators rather than twelve; e.g., G=<B,B',R,R',D,D'>. In the face turn metric, the result will be nine generators rather than eighteen; e.g., G=<B,B',B2,R,R',R2,D,D',D2>. If you do it that way, GAP will correctly calculate the order of the 2x2x2 group.
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Automorphismgroup of 2x2x2 cube recalculated

Submitted by cubex on Mon, 07/25/2005 - 00:03.
I recalculated AutomorphismGroup(cube) using only 3 generators G=< F,L,D >. Size(cube) where cube is the 2x2x2 cube gives the expected 3674160. Size(AutomorphismGroup(cube)) returns a group with 5 generators and size 7348320. Being curious what the generators of AutomorphismGroup(cube) looked like I used GeneratorsOfGroup(AutomorphismGroup(cube)) and it returned:

[ ^(1,9,35)(3,27,33)(6,43)(8,25,19)(11,30)(14,46,40)(16,22,41)(17,24), ^(3,27)(6,43,17,30,11,24)(8,25)(9,
    35)(14,40)(22,41), ^(1,3,8,6)(9,33,25,17)(11,35,27,19), ^(1,17,41,40)(6,22,46,35)(9,11,16,14),
  ^(6,25,43,16)(8,30,41,11)(17,19,24,22) ]

I'll post a link for the GAP file for the 2x2x2 cube. Of course you are right to suggest using less generators as we generally are not interested in full cube rotations in space, plus the calculation of Automorphism(cube) is much faster using just 3 generators.
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So Size(AutomorphismGroup(cub

Submitted by jaap on Mon, 07/25/2005 - 02:13.
So Size(AutomorphismGroup(cube)) = 2 * Size(cube).

Obviously the factor of Size(cube) is due to the inner automorphisms, i.e. the conjugations.
Now where does that factor 2 come from? Is it reflection?
Why wouldn't it be a factor 6, due to reflection plus the rotation around the URF corner?



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Distance preserving automorphisms

Submitted by jaap on Tue, 07/19/2005 - 04:06.
I don't know how to do that in Gap.

Let S be the set of generators and their inverses:
S={F,L,U,B,R,D,F',L',U',B',R',D'}.
You are asking for all automorphisms that map S onto itself. (S contains all the elements of length 1, and order 4, so its elements must be mapped to each other if distance is to be preserved.)

Clearly every cube rotation/reflection is an automorphism. I think these are the only ones you can have that are not the identity on S. (You'll have to verify this - Hint: opposite faces commute, and hence their images also commute.)

Are there then any automorphisms that are like the identity on S, i.e. that map each generator to itself?
No. Every element of the group is a product of the generators, so once the automorphism is known for the generators, it is known for the whole group.

It took me a while to come up with the above. I found it quite confusing, because I kept thinking that the superflip would be another automorphism, but it isn't.

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If I'm understanding the ques

Submitted by Jerry Bryan on Fri, 07/22/2005 - 11:19.
If I'm understanding the question correctly, I don't think it's the rotations/reflections that are the distance preserving automorphisms. It is only the identity rotation/reflection that preserves the identity permutation, so the rest of the rotations/reflections are not automorphisms at all.

Rather, it's conjugation by the rotations/reflections that are the distance preserving automormphisms. Using the cube-lovers convention that M is the set of rotations/reflections and that m is an element of M, then G^m: x->m'xm is a distance preserving automorphism. G^m is an outer automorphism because m is not in G (except in the trivial case where m is the identity), but m is in the larger group GM which we may inefficiently generate as GM=<G,M> (and note that G is a subgroup of GM).

Jerry Bryan
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conjugation vs relabelling

Submitted by jaap on Mon, 07/25/2005 - 02:06.
"Rather, it's conjugation by the rotations/reflections that are the distance preserving automormphisms."

Yes, you are correct, I should have been more careful.

I think we simply look at the same thing with different points of view.

I tend to think of cube rotations/reflections not just as an operation of the cube in 3d space, but actually as a permanent relabelling of the colours/faces/generators/pieces.

Such relabelling is indeed really a conjugation with the 3d rotation/reflection.



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