From BRYAN@wvnvm.wvnet.edu Fri Dec 9 22:20:45 1994 Return-Path: Received: from WVNVM.WVNET.EDU by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA14904; Fri, 9 Dec 94 22:20:45 EST Message-Id: <9412100320.AA14904@life.ai.mit.edu> Received: from WVNVM.WVNET.EDU by WVNVM.WVNET.EDU (IBM VM SMTP V2R2) with BSMTP id 4266; Fri, 09 Dec 94 22:20:47 EST Received: from WVNVM.WVNET.EDU (NJE origin BRYAN@WVNVM) by WVNVM.WVNET.EDU (LMail V1.2a/1.8a) with BSMTP id 5379; Fri, 9 Dec 1994 22:20:47 -0500 X-Acknowledge-To: Date: Fri, 9 Dec 1994 22:20:42 -0500 (EST) From: "Jerry Bryan" To: "Cube Lovers List" Subject: Normal Subgroup Question On 12/07/94 at 20:45:00 Martin Schoenert said: >Unfortunately C is *not* a normal subgroup of CG, and therefore CG/C is >*not* a group. If we want to apply group theory, we need a better model. >I argue that G is indeed a good model for the 3x3x3 cube. I responded at great length, showed a group for CG/C, and concluded as follows. >I guess this must mean that C[C], C[E], and C[C,E] are all normal >subgroups of their respective CG's, but that C[C,F], C[E,F], and >C[C,E,F] are not. That should not be surprising. Having the >Face-centers there only as a frame of reference and never moving >is not the same as having them there and really moving (as when you >rotate the entire cube). This just *has* to be wrong. I just don't see any way that any of the flavors of C are a normal subgroup of their respective flavors of CG. The presence or absence of the Face-centers can't have anything to do with it. I was jumping to the conclusion that since I found a group for some of the flavors of CG/C, that therefore the respective C's must be normal. I have reread my note, and it still looks to me like I found groups for all the CG/C's I discussed. I would invite instruction and correction from any of you group theory experts out there, but here is the way it looks to me. Using G and H generically for a group and subgroup (not necessarily cubes at all), G/H is a group if H is a normal subgroup of G, under the "natural" operation {Xh} * {Yh} = {(XY)h} (where {Xh} etc. denotes all h in H.) Coset notation would be (xH)(yH)=(xy)H. Under these circumstances, G/H is the factor group of H in G. My group operation on the cosets not the "natural" operation. It gets around the fact that C is not normal by picking specific rather than arbitrary elements of the cosets in order to perform the group operation, namely a picking specific element which fixes the same cubie for all cosets. I guess this means that CG/C is not the factor group of C in CG (such a thing being impossible), but by golly it still looks like a group to me under the "unnatural" operation ?? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 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