From:

~~~ Subject:

I said:

In fact, I have now verified with a quick search program that

all M-conjugates in the slice group are also C-conjugates. Hence,

there are 50 C-conjugate classes in slice, just as there are

50 M-conjugate classes.

In retrospect, I don't think the search program was necessary....

On 05/23/95 at 13:11:27 hoey@AIC.NRL.Navy.Mil said:

and (Jerry) continues with an argument that did not convince me, but the

following does:

I think I can both greatly simplify and greatly strengthen the

argument that did not convince Dan. My argument is based on the idea

(copied from _Symmetry and Local Maxima_) that M-conjugation can be

viewed as a permutation on Q, the set of twelve quarter turns.

Call the six clockwise quarter turns Q+ and the six counter-clockwise

quarter turns Q-. We can observe that the 24 rotations in M all

map Q+ to Q+ and map Q- to Q-, and that the 24 reflections in M all

map Q+ to Q- and map Q- to Q+. We also note that in particular,

the central inversion v is a reflection.

Suppose X and Y are M-conjugates in <slice> with Y=m'Xm for some fixed

m in M. Write X as pairs of quarter turns (each pair is a slice), and

write Y as pairs of quarter turns which are respective M-conjugates

(via the fixed permutation m) of the quarter turns in X. If the

respective quarter turns have been mapped Q+ to Q+ and Q- to Q-, then

m is a rotation and we are done.

Otherwise, commute the halves of each slice in Y. We first note that

so commuting is the identity on Y. We also note that so commuting is

equivalent to performing the permutation operation v on Q, and is

therefore equivalent to performing v-conjugation on Y. (In passing,

we see that this effectively proves Dan's first point, namely that

X=v'Xv for all X in <slice>. Given that, I would shorten the rest of

Dan's argument by saying Y=m'Xm=v'('m'Xm)v=v'm'Xmv, and noting that

either m or mv is a rotation).

But having started with the "commuting the halves of slices" argument,

I would continue as follows. Having commuted the halves of the slices,

we still have an M-conjugate (and still the same M-conjugate)

because commuting is equivalent to v-conjugation, v is in M, and

v-conjugation is the identity in <slice>. Finally, having commuted

the halves of the slices, we are now mapping Q+ to Q+ and Q- to Q-,

so we have a rotation.

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