From BRYAN@wvnvm.wvnet.edu Wed Oct 18 20:56:25 1995 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA09250; Wed, 18 Oct 95 20:56:25 EDT Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R3) with BSMTP id 5392; Wed, 18 Oct 95 20:56:03 EDT Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 1451; Wed, 18 Oct 1995 20:56:04 -0400 Message-Id: Date: Wed, 18 Oct 1995 20:56:03 -0400 (EDT) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Re: Positions 8q from Start, 9q from B, Five Generators In-Reply-To: Message of 10/17/95 at 19:44:40 from BRYAN@wvnvm.wvnet.edu I can add a bit of additional information. The 16 positions 8q from Start and 9q from B can be reduced to 4 positions unique up to Q2-conjugacy. As I have discussed before, it is still difficult to claim that the 4 positions are really "different" without further analysis because of the possibility that the positions are variations within a commuting subsequence of moves. I don't really have a Q2-conjugacy program. It would be easy to make one, but I don't have time so I used my M-conjugacy program. Recall that Q2={i,b,bb,bbb,rrv,rrbv,ttv,ttbv}, where b, r, and t are whole cube rotations of the Back, Right, and Top faces, respectively, and v is the central inversion. For 12 of the 16 positions X the program reports Symm(X)={i}, which is to say m'Xm is not equal X for any m in M except the identity. Obviously, the same is true for all m in Q2 since Q2 is a subgroup of M. We have |Q2|=8, so |{m'Xm | m in Q2}=6. Therefore, the 12 positions for which Symm(X)={i} form two Q2-conjugacy classes. Using the M-conjugacy program for the other 4 positions is trickier, but only slightly so. For the other 4 positions, the M-conjugacy program reports Symm(X)=HX, where HX={i,bb,rr,tt,v,bbv,rrv,ttv}. But HX is not a subgroup of Q2, and what we need is sort of "Symm(X) with respect to Q2", which I will call Symm(X/Q2). (A better notation is probably available). It is easy to see that Symm(X/Q2)=(Symm(X) intersect Q2), and we have (HX intersect Q2)={i,ttv,bb,rrv}. This subgroup is called HQ2 in Dan's taxonomy. We have |Q2|=8 and |HQ2|=4, so |{m'Xm | m in HQ2}|=2 when Symm(X/Q2)=HQ2. Therefore, the last 4 positions form two Q2-conjugacy classes. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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