   Date: Wed, 18 Oct 95 20:56:03 -0400   From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~  Subject: Re: Positions 8q from Start, 9q from B, Five Generators

8q from Start and 9q from B can be reduced to 4 positions
unique up to Q2-conjugacy. As I have discussed before,
it is still difficult to claim that the 4 positions are really
"different" without further analysis because of the possibility
that the positions are variations within a commuting subsequence
of moves.

I don't really have a Q2-conjugacy program. It would be easy to
make one, but I don't have time so I used my M-conjugacy program.
Recall that Q2={i,b,bb,bbb,rrv,rrbv,ttv,ttbv}, where b, r, and
t are whole cube rotations of the Back, Right, and Top faces,
respectively, and v is the central inversion. For 12 of the
16 positions X the program reports Symm(X)={i}, which is to say
m'Xm is not equal X for any m in M except the identity. Obviously,
the same is true for all m in Q2 since Q2 is a subgroup of M. We
have |Q2|=8, so |{m'Xm | m in Q2}=6. Therefore, the 12 positions
for which Symm(X)={i} form two Q2-conjugacy classes.

Using the M-conjugacy program for the other 4 positions is trickier,
but only slightly so. For the other 4 positions, the M-conjugacy
program reports Symm(X)=HX, where HX={i,bb,rr,tt,v,bbv,rrv,ttv}.
But HX is not a subgroup of Q2, and what we need is sort of
"Symm(X) with respect to Q2", which I will call Symm(X/Q2).
(A better notation is probably available). It is easy to see
that Symm(X/Q2)=(Symm(X) intersect Q2), and we have
(HX intersect Q2)={i,ttv,bb,rrv}. This subgroup is called
HQ2 in Dan's taxonomy.

```We have |Q2|=8 and |HQ2|=4, so |{m'Xm | m in HQ2}|=2
when Symm(X/Q2)=HQ2.  Therefore, the last 4 positions form
two Q2-conjugacy classes.

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