From @mail.uunet.ca:mark.longridge@canrem.com Sun Sep 24 22:25:13 1995 Return-Path: <@mail.uunet.ca:mark.longridge@canrem.com> Received: from seraph.uunet.ca (uunet.ca) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA16210; Sun, 24 Sep 95 22:25:13 EDT Received: from portnoy.canrem.com ([198.133.42.17]) by mail.uunet.ca with SMTP id <250313-1>; Sun, 24 Sep 1995 22:27:40 -0400 Received: from canrem.com by portnoy.canrem.com (4.1/SMI-4.1) id AA03338; Sun, 24 Sep 95 22:20:55 EDT Received: by canrem.com (PCB-UUCP 1.1f) id 1F62D8; Sun, 24 Sep 95 22:13:27 -0500 To: cube-lovers@life.ai.mit.edu Reply-To: CRSO.Cube@canrem.com Sender: CRSO.Cube@canrem.com Subject: Least Commutative Element From: mark.longridge@canrem.com (Mark Longridge) Message-Id: <60.1242.5834.0C1F62D8@canrem.com> Date: Sun, 24 Sep 1995 22:56:00 -0400 Organization: CRS Online (Toronto, Ontario) # The 2nd most commutative element of the cube group # It is the position of 7 clockwise and 1 counter-clockwise twists # This commutes with 1 out of every 8 elements of the cube commuter := ( 1,35, 9)( 3,27,33)( 6,11,17)( 8,19,25)(24,43,30)(32,48,38) (14,40,46)(16,22,41);; # The least commutative element of the cube group ( I think! ) # This commutes with 1 out of every 450,541,700,775,936,000 # or approximately 1 out of every 4.5 * 10^17 patterns least := ( 1, 6,32,19, 3,41,24,46)( 2, 4,13,42,29, 7,31,15,39,37,26,21) ( 5,28,34,10,20,23,36,18,45,44,47,12)( 8,33,16,30,40, 9,17,38) (11,48,25,27,22,43,14,35);; > after thinking about it, i realized that > > corners: (8) edges: (12) > > commutes with even fewer elements. again, elements with > this cycle structure split into two conjugacy classes. > > mike With GAP we must deal with permutations of cube facelets, and that is why the permutation 'least' has 3 sets of 8 numbers and 2 sets of 12 numbers. Moreover, as I'm sure Mike will appreciate, the least commutative element I've found is has a 8-cycle of corners and a 12-cycle of edges. Size (Centralizer (cube, least)); 96 -> Mark <-