From mreid@ptc.com Tue Mar 7 10:11:22 1995 Return-Path: Received: from ptc.com (poster.ptc.com) by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA04717; Tue, 7 Mar 95 10:11:22 EST Received: from ducie by ptc.com (5.0/SMI-SVR4-NN) id AA07245; Tue, 7 Mar 95 10:09:26 EST Received: by ducie (1.38.193.4/sendmail.28-May-87) id AA25628; Tue, 7 Mar 1995 10:25:12 -0500 Date: Tue, 7 Mar 1995 10:25:12 -0500 From: mreid@ptc.com (michael reid) Message-Id: <9503071525.AA25628@ducie> To: cube-lovers@ai.mit.edu, mark.longridge@canrem.com Subject: New GAP insights Content-Length: 1426 mark writes [ ... ] > The following command shows that pons is at the centre of slice group: > Size (Centralizer (slice, pons)) = 768 this is not hard to see. the center of the slice group has order 32. if we hold the corners fixed, then these central elements are exactly those for which the six face-centers are correct. [ ... ] > Of course, now that I have answered my old questions, I must > formulate new ones.... > > A) What is the next most commutative element (the pancentre?) > after the 12-flip? > B) What is the least commutative element (the anticentre?) of > the cube group? i'm sure that GAP can do these. you're interested in knowing about the orders of centralizers of various elements. for an element g in a group G , we have |G| / |Z(g)| = number of conjugates of g . this is because the cosets G / Z(g) are in one-to-one correspondence with the conjugates of g. of course, this doesn't help much unless we know about conjugacy classes in G. in the case of the cube group, however, conjugacy classes are easy to understand. they are (almost) completely described by cycle structure. (some cycle structures have two conjugacy classes.) there are many different possible cycle structures, but for each it should be easy to count the number of elements with that cycle structure (and also to tell whether they comprise a single conjugacy class or split into two). mike