Date: Tue, 08 Nov 94 17:59:31 -0500 (EST)
From: Dan Hoey <hoey@aic.nrl.navy.mil >
Subject: Re: The real size of cube space

Wow, I didn't realize this sort of calculation had been automated.

Martin.Schoenert@math.rwth-aachen.de writes:

The way I view this is as follows. The entire cube group C is a
permutation group group on 6*9 points, generated by the six face
turns U, D, L, R, F, B; the three middle slice turns M_U, M_L,
M_F; and the reflection S. This group has a subgroup M of
symmetries of the cube (of order 48), generated by U M_U D',
L M_L R', F M_F B', and S. Another subgroup is G, generated by
the six face turns, which has index 48 in G. G is a normal
^
divisor of C, G is the semidirect product of M and G. The same is
^
true for GE and GC.

I think two of those G's are supposed to be C's, right? What is the
difference between a direct product and a semidirect product?

... [conjugation by] M fixes the set of the generators of G and
their inverses. M is fact the largest subgroup of the outer
autmorphism group with this property, which makes it rather
important.

In a 1983 Cubic Circular article (of which I know only Stan Isaacs's
summary) David Singmaster observed that the group is larger for larger
cubes, provided we work what I call the ``theoretical invisible
group''. That is, we solve not only the surface of the cube, but the
hypothetical interior (n-2)^3 cube, and all the smaller (n-2k)^3 cubes
as well. I blithered at length about this in my article of 1 June
1983 archived (I think I've got it right this time) at
<ftp://ftp.ai.mit.edu/pub/cube-lovers/cube-mail-5.gz>.

The idea is that a mapping called evisceration allows us to permute
the layers of the cube. On the 4x4x4 cube, this for instance allows
us to exchange each inner slab with its adjacent outer slab. It also
allows us to conjugate each inner slab move by central inversion,
while leaving the outer slab moves alone. In general, evisceration of
a d-dimensional cube by f maps each feature (cubie, colortab, or
face-center arrow) at coordinates (x[1],x[2],...,x[d]) to
(f(x[1]),f(x[2]),...,f(x[d])), where f is a permutation of the
intervals between the cleavage coordinates of the cube. I believe
that if f commutes with the central inversion, then conjugation by
evisceration is an outer automorphism of the Rubik's cube group. (I
think I have proved this for d=3, and I think the proof in higher
dimensions should not be difficult given the right notation.)

The group of all eviscerations includes the central inversion; we can
of course augment it by the rotation group in d-space. Is this the
maximum outer automorphism group that respects generators of the
Rubik's cube? For this we take the generators to be turns of slabs
between adjacent cleavage planes. (Turns are direct d-1-dimensional
isometries.)

I was already familiar with this augmented symmetry group because it
also induces automorphisms on d-dimensional tic-tac-toe. (In fact, it
may be the maximal automorphism group on all tic-tac-toe boards of
side greater than two. I know it's been proven for 4^3, but I don't
know of any larger results). Do you know anything more about this
group, like whether it has been named or studied?

Since G's structure is very similar to a symmetric group (or more
accurately the direct product of two symmetric groups), it allows
to describe the centralizer of an element in G. The more a group
differs from a symmetric group the less this analysis helps (for
those that know what I'm talking about: the more a group differs
from the symmetric group, the worse a backtrack computation using
cycle structure analysis is).

But no, G's structure is actually similar to the direct product of two
_wreathed_ symmetric groups. Does this interfere with the
backtracking as much as it interferes with my manual analysis? Do you
know of any good treatments of finding centralizers of outer
automorphisms of wreath products? In particular, I would very much
like to know under what conditions the centralizer of the wreath
product fails to cover the centralizer of the permutation factor, as
we saw with the corners.

As for when I wrote

```M class             Edge         Corner       Corner times edge
(class size)      F.P.          F.P.             / (96*class size)
^^^^^^^^^^^^^^^^^^^^^^
```

That's not a typo. I was just saying that column 4 is equal to column
2 times column 3, divided by column 1, divided by 96. Perhaps I
should have factored column 1 out of columns 2 and 3 first to avoid
this confusion.

```gap-3.4 -b -g 4m
gap> Sum( ConjugacyClasses( M ),
>         c -> Size( Centralizer(G,Representative(c)) ) / 48 * Size(c) );
901083404981813616
```

Well, call me John Henry. Say, do you have gap libraries for other
magic polyhedra? For higher-dimensional magic?

Dan Hoey
Hoey@AIC.NRL.Navy.MIl