On 05/11/95 at 17:57:14 mreid@ptc.com said:

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.)

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.

If I understand your definition of "P" correctly, the same group is
called X1 in Dan's taxonomy. X2 similarly preserves the F-B
axis, and X3 similarly preserves the R-L axis. Hence, there are
three conjugate subgroups of G which preserve a major axis, and
each contains 128 elements: there are 128 X1-symmetric positions,
128 X2 symmetric positions, and 128 X3 symmetric positions.

I was bothered by your statement that there were 128 P-symmetric
positions at first because I was equating "P-symmetric" with
"X-symmetric" rather than with "X1-symmetric". There should be
376 X-symmetric positions -- 124 that are X1-symmetric and not
M-symmetric, 124 that are X2-symmetric and not M-symmetric, 124
that are X3-symmetric and not M-symmetric, and 4 that are M-symmetric.

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