From cube-lovers-errors@curry.epilogue.com Tue Aug 6 14:37:19 1996 Return-Path: cube-lovers-errors@curry.epilogue.com Received: from curry.epilogue.com (localhost [127.0.0.1]) by curry.epilogue.com (8.6.12/8.6.12) with SMTP id OAA24897; Tue, 6 Aug 1996 14:37:18 -0400 Precedence: bulk Errors-To: cube-lovers-errors@curry.epilogue.com Date: Tue, 06 Aug 1996 08:58:23 -0500 (EST) From: Jerry Bryan Subject: Commuting Sets To: Cube-Lovers Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Content-Transfer-Encoding: 7BIT If X and Y are sets of permutations, we define XY to be the set {xy | x in X and y in Y}. In my various search programs, I have encountered a number of cases where we have XY=YX, even though we do not in general have xy=yx. For example, let Q[n] be the set of all positions which are n quarter turns from Start. My standard breadth first search is essentially Q[n+1] = Q[n]Q[1] - Q[n-1]. But we could just as well say Q[n+1] = Q[1]Q[n] - Q[n-1] because Q[n]Q[1] and Q[1]Q[n] are the same set. I have been wondering, what are the necessary and sufficient conditions for XY = YX? Note that X and Y are not necessarily groups. I really don't know the answer, and I wondered if anybody out there does. I have some suspicions it has something to do with conjugacy. In all the cases I have worked with, it it the case that if x in X and y in Y, then all the K-conjugates of x are also in X and all the K-conjugates of y are also in Y -- where K is usually M, the set of 48 rotations and reflections of the cube. For other searches such as , K is the symmetry group associated with the group being searched. It is trivial to make an X and Y that don't "commute" in this matter. That is, pick x and y that don't commute and have sets X and Y containing only the single elements x and y, respectively. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) jbryan@pstcc.cc.tn.us Pellissippi State (423) 539-7127 10915 Hardin Valley Road (423) 694-6435 (fax) P.O. Box 22990 Knoxville, TN 37933-0990