.
So I compute the image of a point under the permutation ( ) by
first computing the image of

under and then computing the image
of that point under . For this order of multiplication it is usual
to write

^ for the image of a point

under a permutation
(instead of writing (

), which would be better for the other order).
For this order of multiplication we must define conjugation of by
as ^ := ^-1 (instead of ^ := ^-1).
In this notation, it is certainly true that d(,) = d(,).
This is because each process that transforms to the state ,
will also transform to , and likewise each process that
transforms to will also transform to .
In a certain sense we don't need this though. What you are looking
for is a process

that effects the state , i.e.,

= .
If such a process of length 22 exists, then there exist two processes
and of length 11, such that = .
We can rewrite this as = ^-1. Let T be the set of
elements reachable from by a process of length 11. Note T^-1 = T.
So we see that if there is a process of length 22 effecting ,
then the intersection ( T) ( T) must be nonempty.
As mentioned above, you can interpret the set ( T) as the set
of elements at distance 11 from , but you don't have to.
Now for the superflip you even have d(,) = d(,),
since = because the central commutes with every .
Put differently this means that ( T) = (T ), i.e., instead
of multiplying each element of T from the left by , you can instead
multiply each element from the right.
Have a nice day.
Martin.
-- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .-
Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551
Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany