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Date: Fri, 01 May 1998 22:35:29 -0400
From: Mark Longridge
Reply-To: cubeman@idirect.com
To: Dan Hoey
Cc: cube-lovers@ai.mit.edu
Subject: Re: Square like groups
References: <9805012356.AA16835@sun28.aic.nrl.navy.mil>
Dan Hoey wrote:
>
> Andrew Walker asks:
>
> > Does anyone have any information on patterns where each
> > face only contains opposite colours, but are not in the square
> > group?
>
> We may call this the "pseudosquare" group P. It consists of
> orientation-preserving permutations that operate separately on the
> three equatorial quadruples of edge cubies and the two tetrahedra of
> corner cubies, and for which the total permutation parity is even. So
> Size(P) = 4!^5 / 2 = 3981312.
>
> > L' R U2 L R' may be an example.
R2 F2 R2 U2 R2 F2 R2 U2 F2
>
> No, that's in the square group, says GAP. Also, Mark Longridge
> noticed (8 Aug 1993) that the square group is mapped to itself under
> conjugation by an antislice (though I don't recall a proof--is there
> an easy one?). Your position is (L R)' R2 T2 R2 (L R), so this result
> would apply. Does anyone have a square process for it?
I almost forgot about all that info back in 1993!
But I hardly think a proof is necessary. After the moves (L' R) all the
following moves are in the square's group. Then we are just doing
the inverse of (L` R) at the end. Not very rigourous, but...
I'll search for a counter-example.
-> Mark <-