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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.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU