From:

~~~ Subject:

On 12/07/94 at 20:45:00 Martin Schoenert said:

Unfortunately C is *not* a normal subgroup of CG, and therefore CG/C is

*not* a group. If we want to apply group theory, we need a better model.

I argue that G is indeed a good model for the 3x3x3 cube.

I responded at great length, showed a group for CG/C, and concluded

as follows.

I guess this must mean that C[C], C[E], and C[C,E] are all normal

subgroups of their respective CG's, but that C[C,F], C[E,F], and

C[C,E,F] are not. That should not be surprising. Having the

Face-centers there only as a frame of reference and never moving

is not the same as having them there and really moving (as when you

rotate the entire cube).

This just *has* to be wrong. I just don't see any way that any

of the flavors of C are a normal subgroup of their respective

flavors of CG. The presence or absence of the Face-centers can't

have anything to do with it. I was jumping to the conclusion that

since I found a group for some of the flavors of CG/C, that therefore

the respective C's must be normal.

I have reread my note, and it still looks to me like I found groups

for all the CG/C's I discussed. I would invite instruction and

correction from any of you group theory experts out there, but here

is the way it looks to me.

Using G and H generically for a group and subgroup (not necessarily

cubes at all), G/H is a group if H is a normal subgroup of G,

under the "natural" operation {Xh} * {Yh} = {(XY)h} (where {Xh}

etc. denotes all h in H.) Coset notation would be (xH)(yH)=(xy)H.

Under these circumstances, G/H is the factor group of H in G.

My group operation on the cosets not the "natural" operation.

It gets around the fact that C is not normal by picking specific

rather than arbitrary elements of the cosets in order to perform

the group operation, namely a picking specific element which fixes

the same cubie for all cosets. I guess this means that CG/C is not

the factor group of C in CG (such a thing being impossible), but

by golly it still looks like a group to me under the "unnatural"

operation ??

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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