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Date: Wed, 19 Aug 98 23:34:34 EDT
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From: Dan Hoey
To: jbryan@pstcc.cc.tn.us
Cc: reid@math.brown.edu, cube-lovers@ai.mit.edu
In-Reply-To: (message from Jerry
Bryan on Wed, 19 Aug 1998 17:09:54 -0400 (Eastern Daylight Time))
Subject: Re: minimal maneuvers for X symmetric positions
Jerry Bryan writes:
> Positions #63 through #124 are essentially the first 62
> positions composed with superflip. I had never noticed it, and
> I don't *think* it has been described on the list, but for every
> symmetry group, half of the corresponding positions can be
> described as "basic" positions and the other half can be
> described as the basic positions composed with superflip. That
> is, if Symm(x)=K, then Symm(xf)=Symm(fx)=K, where x is any
> position and f is the superflip.
This is easy to see if we consider that Symm(x) is the set of all
m in M that commute with x, because
m' x m = x if and only if x m = m x.
At some times since 1981 I've wondered whether symmetry discussions
are better done with commutativity rather than conjugacy.
So if c is any element of the center of G* -- i.e., c commutes
with all elements of M and G -- then Symm(x)=Symm(x c). As is
well known to cube-lovers, the center of the usual cube group consists
of the identity and the superflip. In the supergroup, we may also
compose these with Big Ben (all face centers rotated 90 degrees) and
Noon (Big Ben squared).
Dan Hoey
Hoey@AIC.NRL.Navy.Mil