From BRYAN@wvnvm.wvnet.edu Thu May 18 11:30:01 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA00211; Thu, 18 May 95 11:30:01 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 0025; Thu, 18 May 95 11:28:37 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 0519; Thu, 18 May 1995 11:28:37 -0400 Message-Id: Date: Thu, 18 May 1995 11:28:36 -0400 (EDT) From: "Jerry Bryan" To: , "Cube Lovers List" Subject: Re: more on the slice group In-Reply-To: Message of 05/11/95 at 17:57:14 from mreid@ptc.com 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