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I feel like I am pestering the list to death with corrections.

I still believe that the figure that I proposed for the size of

the cube group under M is correct. The first post included

a "correct" but I think unsatisfactory explanation. The second post

improved upon one point that was unsatisfactory in the first post.

Now, let's see if I can get it completely correct.

The size of the corner group under (my version of) M is known.

The size of the edge group under (my version of) M is known as

well. Let C be the size of the corner group, and E be the size

of the edge group. Remember, the elements of the groups are

equivalence classes induced by (my version of) M. Here is an incorrect

formula for G, the size of the entire cube group under (my

version of) M.

G = (C*24) * (E*24) / 2

The division by 2 is introduced to account for parity between the

corner group and the edge group. But the value for G produced by

this formula is only half as big as it should be. The problem is

that M induces equivalence classes based on both rotations and

reflections, not just base on rotations. Hence, we are led to the

following (still incorrect) formula:

G = (C*24*2) * (E*24*2) / 2

As before, the division by 2 takes care of parity between the corner

group and the edge group. In addition, the multiplication by 2 takes

care of reflecting each group. But the value for G produced by this

formula is twice as big as it should be. The problem is that while

any corner rotation can occur with any edge rotation (subject to

parity), you must either reflect both groups, or else reflect neither

group. Thus, we have the following (correct) formula:

G = ((C*24) * (E*24) / 2) * 2

The division by 2 takes care of parity between the groups, and the

multiplication by 2 takes care of reflection of the two groups

as a unit. If we wish, we can cancel the multiplication and the

division to yield

G = (C*24) * (E*24)

This is the same formula I originally posted, and I did say in the

original post that the division by 2 cancelled out. However, I

think that this post provides a better explanation of the

cancellation than did the original post.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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?