   Date: Thu, 22 Jun 95 03:43:59 -0400 (EDT)   From: Dan Hoey <hoey@aic.nrl.navy.mil >
~~~ ~~~ Subject: Ways to Calculate the Real Size of Cube Space?

Martin Schoenert says:
> I can't figure out how the brute force computer search works.

and while Jerry Bryan gives one answer, I have another. For you see,
I also ran a brute force computer search for the symmetry classes,
too. And while it agrees with Jerry's answers, mine was a
significantly different algorithm, which I discuss a little a the end
of this message.

It appears to me that Dan and Jim Saxe must have realized all the
important pieces for your new method when they wrote their seminal
``Symmetry and Local Maxima (long message)'' message of 1980/12/14.
As Jerry points out, they did calculate the important values for
9 of the 33 conjugacy classes of subgroups of M (those whose sizes
are a multiple of 12). It is neither clear from their message how
they found those 9 classes (in fact they apparently found all 98
subgroups of M), nor how they computed the numbers of elements of
G that have a specific subgroup of M as symmetry group.

First, to find all the subgoups of M. I represented the elements of M
as a list of permutations on faces, found easily enough by finding the
closure of the generators. Then I did a depth-first search for
subgroups, branching by computing the closure of the current subgroup
with each possible element not in that subgroup, and cutting off the
search on previously-seen subgroups. It's quick, it's dirty, it
works. I found the nine subgroups of order a multiple of 12, as shown
in the Hasse subgroup diagram:

```order 48 . . . M_
/|\\_
/ | \ \_
/  |  \  \_
order 24 . A . H . C   \_
\  |  /      \_
\ | /         \_
\|/            \
order 12 . . . E . . . . .T[1..4]
```

(except we called E "AC" back then). The trick then was to find all
the E-symmetric positions and all the T1-symmetric positions; the
tasks of finding the full symmetry group of such positions and
counting the positions for T2, T3, and T4 were straightforward.

The best tool we had for figuring out symmetric positions was
essentially the one I wrote about in ``The real size of cube space''
on 4 Nov 94. For a subgroup J of M, if a position g is J-symmetric,
then g must commute with each operation m in J. Recall:

```The fundamental principle we use in finding whether g commutes with
m can be found by examining the cycles of m.  Suppose m permutes a
cycle (c1,c2,...,c[k-1],ck), so that
c2=m(c1), c3=m(c2), ..., ck=m(c[k-1]), and c1=m(ck).
For g to commute with m, we have
g(c2)=m(g(c1)), g(c3)=m(g(c2)), ..., g(ck)=m(g(c[k-1])),
and g(c1)=m(g(ck)).
So (g(c1),g(c2),...,g(ck)) is also a cycle of m.  Thus g must map each
k-cycle of m to another k-cycle of m, and in the same order.

The orientation question was a lot more difficult, so we ran through a
bunch of little results.  The following is a cleaned-up sample of the
sort of arguments, as I remember them.  In it FRD is an unoriented
corner cubie/cubicle and FRD.D is its down-facing color-tab/facicle.
================================================================
Lemma 1: Suppose X and Y are corners, and m is in C, m(X)=Y.
Suppose g(X)=X and g(Y)=Y, and g commutes with m,
Then g applies the same twist to corners X and Y.
Proof: Let TX,TY be the clockwise 120-degree rotation of corners
cubies X and Y, respectively.  Then m(TX(.))=TY(m(.)), as
can be seen by the fact that we could apply a twist to X in
place (TX) before moving it to Y with m, or we can perform
the same twist on Y (TY) after moving it.  So if
g(X)=TX^k(X), then
g(Y)=g(m(X))=m(g(X))=m(TX^k(X))=TY^k(m(X))=TY^k(Y).
performing the same twist on Y as on X,               QED.
```
```Lemma 2: If g is E-symmetric, then each corner cubie remains in
its home cubicle (not considering orientation).
Proof: Supposing otherwise, take a moved cubie (without loss of
generality) to be FRD, and suppose (w.l.o.g.) it moves to
one of locations FDL, FLT, or BTL.
Case 1.  If g(FRD)=FDL, consider operation m to be the
m(FRD)=FRD, m(FDL)=FTR.  So
g(FRD)=FDL.
Case 2.  If g(FRD)=FLT, then take m as in case 1; m(FLT)=BRT,
so g(FRD)=g(m(FRD))=m(g(FRD))=m(FLT)=BRT,
Case 3.  If g(FRD)=BTL, then g(FRD.F) is BTL.B, BTL.T, or
BTL.L.
Case 3a. If g(FRD.F)=BTL.B, then g(FRD.R)=BTL.T, by clockwise
and g(FRD.R)=g(m(FRD.F))=m(g(FRD.F))=m(BTL.B)=BTL.L
Cases 3b and 3c work the same way.
The contradictions establish that FRD does not move,  QED.
```

Lemma 3: If g is E-symmetric, then the twists of the four corner
cubies FRD, FLT, BLD, and BRT agree with each other, and
and the other four also agree with each other.
Proof: For any two of the four corners (e.g. FRD, FLT), there is
a 120-degree rotation in E taking one to the other
(e.g. the rotation about FTR). Lemma 1 applies
immediately to show the twists agree, QED.

Lemma 4. If g is E-symmetric, then the corner cubies are all
solved, or are rotated alternately in opposite
directions.
Proof: From Lemma 2, all the cubies are in their home cubicles.
If one of the sets from Lemma 3 is twisted, their total
twist is the twist of a single cubie (since there are four
of them) and so must be counteracted by having the other
set twisted in the opposite direction, which is alternate
corners twisted oppositely; otherwise all corner are
solved, QED.

```    Lemma 5. Let T1 refer to the group that fixes the FRD-BTL axis.
Then any T1-symmetric position g must keep the FRD and
BTL cubies in their solved position, and rotated by the
same amount.
Proof: Let m be the 120-degree rotation about FRD, which is in T1.
Since FRD and BTL are the only 1-orbits of m, they are kept
in place or swapped.  From the proof of Lemma 2 (which uses
the same m) they cannot be swapped.  Otherwise, the two
cubies are kept in place, and 180 degree rotation about the
FL-BR axis, also in T1 fulfills the requirements of Lemma 1
to show they will both be rotated the same amount,    QED.
================================================================
That's about all I feel like remembering and formalizing right now.
As you can see, it's long, mechanical, and boring.  That's why we
never got around to writing it all down.
```

Early last year I wrote a computer program to find *all* the symmetry
groups of *all* the positions. The first part did the corners and
edges separately, counting the number of positions for each symmetry
group and permutation parity. For each of the 8! or 12! permutations,
I checked to see if the permutation commuted with some nontrivial
operation of M; if not, I just counted the appropriate number of
I-symmetric positions. Otherwise I applied orientations and counted
up the symmetry groups of each possible orientation. (It could
probably have been made to go faster, by cutting off partial
permutation or orientation generation as soon as all the non-trivial
operations were ruled out.)

In the above counts, I also kept track each time of whether the
permutation was even or odd. Then after I had the count of even and
odd permutations for corners and edges in each symmetry group, I had a
program that intersected the symmetry groups, and for each pair of
subgroups J,K, and each parity P, I added
Corners[J,P] * Edges[K,P] to Whole[J intersect K, P],
with some fancy footwork so I only needed to deal with conjugacy
classes for J and K. Then for each K, the number of whole K-symmetric
positions was Whole[K,Even]+Whole[K,Odd].

The program finished about the time I realized the application of the
Polya-Burnside theorem. Then for most of the year, I put off writing
it all up. I have Jerry to thank for reminding me to get with it, and
for useful comments and discussions on the early drafts.

Dan     