   Date: Thu, 25 May 95 11:50:55 -0400   From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~  Subject: Re: M-conjugacy vs. C-Conjugacy in the Slice group

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.

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Robert G. Bryan (Jerry Bryan)                        (304) 293-5192
Associate Director, WVNET                            (304) 293-5540 fax     