On 11/08/94 at 01:18:00 Martin.Schoenert@math.rwth-aachen.de said:
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 have discussed a similar view of things recently, except that I was
not brave enough to include a reflection in the generators. C is
normally used to denote the set of twenty-four rotations of the
cube (a sub-group of M), so let's call your "entire cube group"
big_G instead. My version of big_G was generated by Q plus the
slice moves (like yours without the reflection), or alternatively
by Q plus C. Your version of big_G is hence the same as the one
I discussed except that you added a reflection. C (the rotations C,
that is) is a sub-group of both versions of big_G. M is a sub-group
of your version of big_G, but not of mine.
Your big_G has the obvious advantage of including M as a sub-group.
Mine has the advantage (?) of being physically realizable on a
real cube. That is, for X in your big_G, rX or Xr (r is a reflection)
is also in your big_G. For X in my big_G, rX or Xr is not in
big_G, and correspondingly a single reflection is not physically
realizable on a real cube. Of course, r'Xr is in big_G in either
case, r being in M. Also, cX and Xc are in either version of big_G
for all c in C.
I tend to think that Singmaster's standard G=<Q> is not what people
think of when they hold a real cube in their hand. Rather, they
tend to think of big_G/C. That is, the cosets of C in big_G are
common sensically considered to be equivalent because rotating
a real cube in space is "doing nothing". Also, for my version
of big_G we have |big_G/C| = |G|.
For either version of big_G, we have to re-interpret
parity arguments slightly. In Singmaster's G=<Q>, we say that
even corners occur only with even edges and vice versa. In
big_G, a face quarter-turn is odd on the corners and edges, and
a slice quarter-turn is odd on the edges and on the centers.
Hence, you can have odd corners with even edges and vice versa,
but only if the centers are simultaneously odd. Therefore, the
rules concerning which configurations of edges and corners can
occur together are really preserved, even in big_G.
Finally, neither version of big_G is as big as you can go. That is,
neither of them includes Singmaster's Supergroup, where different
orientations of the otherwise fixed face centers are considered.
Also, neither one of them considers Dan Hoey's Eccentric Slabism,
wherein invisible inner cubes are considered.
Note that the elements of M are also a autmorphisms of the Cayley
graph. That means that elements of M respects the length of operations.
That is if g_1 and g_2 are elements of G that are in one conjugacy class
under M, then the lenght of the shortest process effecting them is equal.
This follows from the fact that 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.
This of course is the basis for the large searches I have been able
to perform using M-conjugate classes. The only trouble is, I don't
even know what a Cayley graph is (but I am working on it), the last
course I took in group theory being 25 years ago.
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