On 13 Feb 1994, I proposed a way to model centerless cubes which would
(in Dan Hoey's words) retain the symmetrical nature of the problem.
I need to post a partial correction/retraction.
The conventional model for centerless cubes loses the symmetrical
nature of the problem. For example, for a corners-only cube, seven
cubies are modeled rather than eight, and for an edges-only cube,
eleven cubies are modeled rather than twelve.
My proposal in February was to use cosets of the form xC to model
centerless cubes, where x is a cube and where C is the set of
twenty-four whole cube rotations. This proposal in turn requires
an interpretation of C such that C is a subset of G, the entire cube
C is a group, but normally it is not considered to be a subset
of G, hence it is not normally considered to be a subgroup of G.
That is, C moves the centers of the faces, but G does not. The
required interpretation is obtained by removing the centers of each
face, and defining rotational orientation by convention so that the
cube is solved only when the Up color is Up, the Front Color is
Front, and so forth.
Under this interpretation, C is indeed a subset (and hence a subgroup)
of G. More correctly, C[even] is a subset of G, C is a subset of
GC (the corners only cube), C is a subset of GE (the edges only cube),
and C is a subset of GS, where GS=<Q,S> (Q is the set of quarter turns
and S is the set of slice moves). That is, when you start
talking about C as a subset of G, you have to worry about odd and
even permutations. Hence, you have to say C is a subset of GS or
C[even] is a subset of G in order not to violate parity rules.
All of the above was posted in February, and I am still comfortable
with it. However, I went on to say that GS/C, G/C[even], GC/C, and
GE/C were all groups under the operation xC * yC = (xy)C. I find
that I must retract this claim.
In my note in February, I did not give a proof, but rather appealed
to a proof in Frey and Singmaster's _Handbook of Cubik Math_. I now
find that I mis-applied their proof. In order to show the nature of
the problem, I find it useful to go through an attempted proof and
determine the point at which it fails.
Note that the proposed group elements are not cubes, they are cosets.
We proceed as follows:
1. Associativity: (xC * yC) * zC = (xy)C * zC = ((xy)z)C = (x(yz))C = xC * (yz)C xC * (yC * zC)
Note that the associativity of the proposed group G/C derives
directly from the associativity of G.
2. Identity: we propose that the identity is iCiC * xC = (ix)C = xC xC * iC = (xi)C = xC
Note that the identity of the proposed group G/C derives
directly from the identity i of G. Further note that the
identity iC of the proposed group G/C is C, which is
precisely what you would want for the identity of a centerless
3. Inverse: we propose that (xC)'=x'CxC * x'C = (xx')C = iC x'C * xC = (x'x)C = iC
Note that the inverse of xC in the proposed group G/C derives
from the inverse of x in G.
4. Closure: This is where we have our problem. We require that
if xC * yC = (xy)C, then (xy)C must be a coset of C. But the
representation of xC and yC is not unique. That is, xC=(xd)C,
where d is in C, and yC=(ye)C where e is in C. It is the
case that (x(ye))C = (xy)C, but in general it is not the case
that ((xd)y)C = (xy)C. Hence, we can have xC=(xd)C, but have
it be the case that xC * yC is not equal (xd)C * yC. Hence,
we do not have closure.
Strictly speaking, this same problem afflicts our "proof" for
the inverse, but I deferred discussing the problem until I got
to closure. If the problem is repaired for closure, it is also
repaired for inverses (see the next paragraph for a discussion
of normal subgroups).
Cosets of a subgroup H are said to be normal if xH = Hx for all x.
I was implicitly and incorrectly assuming that C is a normal
subgroup of G, but it is not. For normal subgroups, closure of
coset multiplication is readily proven. Frey and Singmaster's proof
is for normal subgroups only, and I was applying it to C, which is
It is instructive to consider briefly what xC vs. Cx means for cubes.
We can interpret the left coset xC as simply holding a cube in your
hands and rotating it any way you wish in space without performing
The right coset Cx is a little trickier. The cube x
must be pre-multiplied by each c in C to form Cx. If you have cube
x in your hands, there is no obvious thing you can do to form Cx.
The thing that is most intuitive to me personally is to think in
terms of "rotation by color", which is the way I described
pre-multiplication when I first posted some of the results of my
work back in December. That is, think of holding the cube still, but
recoloring it by permuting the colors. The elements of the coset Cx
look the "same" but with the colors permuted. It is not possible to
perform this operation on a real cube (short of pulling off the stickers
and putting them back on), but the operation can be readily performed
on a computer model.
Having said all this, I keep thinking that there must be a way to define
an operation on the cosets xC so that they form a group. However, I
have been unsuccessful in doing so. I would welcome any advice from
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If you don't have time to do it right today, what makes you think you are
going to have time to do it over again tomorrow?