Date: Fri, 09 Dec 94 22:20:42 -0500
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: Normal Subgroup Question
```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