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I welcome Jerry Bryan's <BRYAN%WVNVM.BITNET@mitvma.mit.edu> efforts to

improve the terminology of the groups associated with Rubik's cube.

But there is some additional clarification I think is necessary.

Let G be the standard cube group for the 3x3x3 cube....

Let GC be the corners with centers without edges group....

Let GE be the edges with centers without corners group....

That much will do, mod quibbles about what name is best.

Let G\C be the corners with edges without centers group. I intend

for the notation to indicate G reduced by C, where C is the rotation

group for the cube....Let GC\C be the corners without edges without centers group....

Let GE\C be the edges without corners without centers group....

First, these are not, strictly speaking, groups. Well, you can make

them groups, by defining what the group operation is. But I don't

know any way of doing that without losing the symmetrical nature of

the problem.

Second, I would suggest that G/C, GC/C, and GE/C are more standard

names for these objects. The elements are nominally 24-element sets,

each of which is an equivalence class when two positions are

considered equivalent when they differ by their position with respect

to the corners. The classes are called the cosets of C in G, GC, and

GE, respectively.

Let G\M be the set of M-conjugate classes for G.....

Let GC\M be the set of M-conjugate classes for GC....

Let GE\M be the set of M-conjugate classes for GE....

The partition of a group into conjugacy classes is not at all the same

as the partition into cosets. So I would prefer to use different

symbology, like "\" for conjugacy and "/" for cosets, but....

Recall that B is the function which calculates the canonical form

for a cube under the composed operations of M-conjugation plus

rotation. My programs calculate equivalence classes under B.Let G\B be the set of B-classes for G [ and likewise for GE, GC ].

Well, if you are using "\" for a generic partition into equivalence

classes, then we should really do something like G\Conj(M) for

partitions into conjugacy classes. At least then you can say

G/C=G\Cosets(C).

Then, we have Gx\B=(Gx\C)\M=(Gx\M)\C. In English, we can decompose

B into a multiplication by C and M (in either order).

No, that's _multiplication_ by C and _conjugation_ by M. A good

example of why it's important not to use confusing symbols. M and C

are not at all treated the same, except inasmuch as they are used to

induce partitions into equivalence classes. Say instead that

Gx\B = (Gx/C)\Conj(M) = (Gx\Conj(M))/C.

If I wanted to model GC\C, I would have had to either model only

seven of the cubies, or else modeled all eight but moved only seven

of them. Since what I really wanted was (GC\C)\M, and since what I

had was GC, I had to invent this funny B thing, where GC\B=(GC\C)\M.

If I had been clever enough to model GC\C in the first place, I

never would have had to invent B. Similar comments apply to my

model for the edges.

Well, the part about moving only seven (corner) cubies is the approach

that's been taken before on this list to deal with cubes that don't

have face centers. It has the advantage that the object being treated

is a group. But the problem is that the group is no longer cubically

symmetrical (in some vague sense). This led me, at least, to lose

track of the structure that would allow analysis of M-conjugacy. So I

have to admire your tackling GE as a whole, instead of trying to stick

to GE/C. At first blush, it looks like GE/C is 24 times smaller. But

since GE/C\Conj(M) is almost 48 times smaller still, it's important to

work in GE at least enough to be able to use the conjugation. Which

is beside the point that I'm actually very interested in the structure

of Gx/Conj(M) itself. And that is what I was really getting at in

1984 when I asked about how many positions there really are.

Dan Hoey

Hoey@AIC.NRL.Navy.Mil