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C is the set of twenty-four rotations of the cube. After much

bungling (see my notes of 13 Feb 1994, 23 May 1994, and 19 July 1994),

I showed that the left cosets of C, denoted by xC or {Xc}, form

a group, and that the group is isomorphic to a subgroup of G. I consider

this to be important because I use left cosets of C to model

centerless cubes.

M is the set of forty-eight rotations and reflections of the cube.

I often model the cube with M-conjugate classes of the form {m'Xm}.

Therefore, it seems that I should try to define an operation such that

the M-conjugate classes form a group, and such that the group is

isomorphic to a subgroup of G.

I would like to start by reviewing briefly the results for left cosets.

Two operations were defined:

1.a. {Xc} * {Yc} = {(VW)c} 1.b. V ** W = (VW)

where V and W are representative elements of {Xc} and {Yc}, respectively.

Further, the mapping V <--> {Vc} defines an isomorphism between the

set of left cosets and the operation * on the one hand, and the set of

representative elements and the operation ** on the other hand. Since

the ** operation is simply normal cube multiplication and since the

set of representative elements are a group under **, the set of

representative elements form a subgroup of G.

I tried to define groups without using representative elements and failed.

Not only that, the representative elements had to be selected in a

special way rather than arbitrarily. For example, we could choose as

the representative element of {Xc} the unique element V such that the

ur cubie is positioned properly.

Positioning the ur cubie properly is not the only selection function for

a representative element which will work, but any selection function must

satisfy two criteria in order to work:

A. It must select a representative element based on a property which

is possessed by exactly one element of each coset.B. There must be closure in the sense that if V is the representative

element of {Xc} and W is the representative element of {Yc},

then (VW) must be the representative element of {(VW)c}.

Criterion #B merits some additional discussion. First, it is the

criterion that really proves you have a group. Associativity

for a subset of a group generally follows from the the associativity

of the group. For a finite group, closure for a subset implies

the identity and the complement for the subset, so closure is the key

factor in demonstrating that a set of cubes is a group. Second,

criterion #B will bear directly on our attempt to define a group

operation for the M-conjugate classes.

Suppose we choose not to require criterion #B. We still need to

have closure in order to have a group. We could obtain closure by

brute force as follows:

2.a. {Xc} * {Yc} = {(Repr{(VW)c})c} 2.b. V ** W = Repr{(VW)c}

It is probably a little easier to see what is going on in equation

2.b. than in 2.a., but it is the identical mechanism in both cases.

Suppose we don't have closure. That is, suppose the selection

function operates in such a way that if V is the representative

element of {Xc} and W is the representative element of {Yc} that

(VW) is not necessarily the representative element of {(VW)c}.

We can still find the representative element of {(VW)c} by simply

applying the selection function, which we have done.

Equations 2.a and 2.b define groups, where the left cosets are

a group under * and the representative elements are a group under

**. Furthermore, the mapping V <--> {Vc} defines an isomorphism

between the two groups. But even though the set of representative

elements is a subset of G, and even though they form a group under **,

they are not a subgroup of G. The problem is that the operation

** as defined by equation 2.b. is not the same operation as

standard cube multiplication as it was in equation 1.b.

Now, let's look at M-conjugate classes. By analogy with the left

coset case, there are two possibilities to define a group:

3.a. {m'Xm} * {m'Ym} = {m'(VW)m} 3.b. V ** W = (VW) 4.a. {m'Xm} * {m'Ym} = {m'(Repr{m'(VW)m})m} 4.b. V ** W = Repr{m'(VW)m}

As before, X and Y are any cubes in G, and V and W are the

representative elements of {m'Xm} and {m'Ym}, respectively.

In order to make 3.a. and 3.b. work, we need some characteristic

which can be used by the selection function which possesses the

properties of uniqueness and closure as defined by #A and #B

above. But I can't think of any such property, and I don't think

such a property exists (see below).

4.a and 4.b certainly work. That is, they define operations

* and ** under which the set of M-conjugate classes and the set

of representative elements, respectively, form groups, and the

groups are isomorphic under the mapping V <--> {m'Vm}.

However, the groups fail to be subgroups of G for the same reason

elements of left cosets fail to be subgroups of G under equation 2.b.

Namely, the ** operation is not really the same operation as

normal cube multiplication.

As to the question of whether 3.a. and 3.b. can be made to work,

I think we can prove that they cannot. Suppose the contrary.

That is, suppose that there is some property such that it is

possessed by exactly one element of each M conjugancy class and

such that the normal cube product of two such elements also

possesses the property. Then, it would be the case that the

set of representative elements would be a subgroup of G. But the

number of representative elements is the same as the number of

M conjugate classes, and the number of M conjugate classes is

known not to divide the number of cubes in G evenly. Hence,

the set of representative elements of M-conjugate classes is not

a subgroup of G. Working backwards contrapositively, the desired

property cannot exist.

So, the final result is that the set of M conjugate classes can be

made into a group, and the set of representative elements of the

M conjugate classes can be made into a group. But neither group

is a subgroup of G, nor is either group isomorphic to any

subgroup of G.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU

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?