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Date: Tue, 23 May 95 13:11:27 EDT
From: hoey@aic.nrl.navy.mil
Message-Id: <9505231711.AA26574@sun13.aic.nrl.navy.mil>
To: "Jerry Bryan" <BRYAN@wvnvm.wvnet.edu>, cube-lovers@ai.mit.edu
Subject: M-conjugacy vs. C-Conjugacy in the Slice group

Jerry Bryan tried to communicate about this to cube-lovers, but
there's apparently a technical difficulty.

On 20 May, Jerry said:

>I doubt that Mark's theory about GAP using C-conjugacy for slice
>instead of M-conjugacy is correct.  I already have 50 positions
>to 23 for GAP, and using C-conjugacy would just make my results
>larger.  For example, RL' and R'L are M-conjugate positions,
>but not C-conjugate positions.

I emailed him to note to the contrary that RL' and R'L are indeed
C-conjugates, for example under 180 degree rotation around the F-B
axis.  I did wonder, though, whether that meant that there could be
C-conjugate slice positions that were not M-conjugate.

He emailed me:

> We can observe that R and R' are not C-conjugates, nor are L' and L,
> which suckered me into stating that RL' and R'L are not.  But
> rewrite R'L as LR' since opposite face moves commute.  Now, RL'
> and LR' are clearly C-conjugate.

> 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....

and continues with an argument that did not convince me, but the
following does:

First, consider the central inversion v, which maps each point of the
cube to its diametric opposite.  Conjugation by v maps each face-turn
(e.g. F) with its diametric opposite in opposite sense (B').  Since
these are the pairs that constitute a slice move, and they commute, we
have:

    v' FB' v = (v' F v)  (v' B' v) = B' F = F B',

and similarly for the other slice moves, showing that each slice move
is its own v-conjugate.  This extends to a proof that each position in
the slice group is its own v-conjugate:

    v' s1 s2 ... sn v = (v' s1 v) (v' s2 v) ... (v' sn v)
                      = s1 s2 ... sn.

Suppose that we have two M-conjugate positions X, Y in the slice
group.  So X = m' Y m for some m in M.  If m is in C, then X and Y are
C-conjugate and we are done.  Otherwise take the central inversion v;
we know that mv is in C.  We also know that

    X = v' X v = v' m' Y m v = (mv)' Y (mv).

So X and Y are C-conjugate in this case as well.  QED.

Note: "Being its own v-conjugate" might as well be called "being
v-symmetric".

Dan
Hoey@AIC.NRL.Navy.Mil