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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