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Date: Thu, 11 May 1995 17:57:14 -0400
From: mreid@ptc.com (michael reid)
Message-Id: <9505112157.AA02793@ducie>
To: cube-lovers@ai.mit.edu
Subject: Re: more on the slice group
Content-Length: 1749
jerry writes
> There are other positions with the same symmetry characteristics as
> the 4-spot. That is, there are other positions for which the
> symmetry group contains sixteen elements. There are only three subgroups
> of M containing sixteen elements, and the three subgroups are M conjugate.
these subgroups are the 2-sylow subgroups of M. one of sylow's
theorems states that any two p-sylow subgroups are conjugate.
one of these subgroups is the group of symmetries that preserve
the U-D axis. call this subgroup "P". (this is also the group
of symmetries of the intermediate subgroup of kociemba's algorithm.)
jerry asks about P-symmetric positions. coincidentally, i happened
to investigate these a few weeks back, and here's what i found:
(i calculated by hand, so i'd be grateful for any confirmation.)
there are 128 P-symmetric positions, 4 of which are M-symmetric.
they form a subgroup of the cube group (of course) which is
isomorphic to a direct product of 7 copies of C_2. in particular,
each such position has order 2 (or 1) as a group element. thus,
the answer to jerry's question
> Call the 4-spot with all edges flipped t. Then, we certainly have
> t'=t. Is this true of all positions whose symmetry group contains
> sixteen elements?
is "yes". for what it's worth, this group of 128 positions can be
generated by the seven elements
superflip
pons asinorum
four spots
slice squared ( U2 D2 )
eight flip ( FB UD RL FB UD RL )
four pluses ( R2 F2 R2 U'D F2 R2 F2 UD' )
four swapped corner pairs ( D' B2 U'D F2 U2 L2 B2 L2 B2 U2 L2 F2 U )
however, these positions are not all locally maximal; for instance
U2 D2 is not.
mike