From scotth@ssd.intel.com Fri Oct 20 19:17:28 1995 Return-Path: Received: from SSD.intel.com by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA23016; Fri, 20 Oct 95 19:17:28 EDT Received: from inchgower.ssd.intel.com by SSD.intel.com (4.1/SMI-4.1) id AA12613; Fri, 20 Oct 95 16:17:06 PDT Message-Id: <9510202317.AA12613@SSD.intel.com> To: Michiel Boland Cc: cube-lovers@ai.mit.edu Subject: Re: Embedding G in a symmetrical group In-Reply-To: Your message of "Fri, 20 Oct 95 02:41:55 BST." <199510200141.CAA03659@wn1.sci.kun.nl> Date: Fri, 20 Oct 95 16:17:04 -0700 From: Scott Huddleston >It is clear that the group G of the cube (the one with >4.3252x10^19 elements) can be embedded in a >symmetrical group, e.g. S_48, since each move of the cube can be >seen as a permutation of 48 objects. Hence, there is a smallest >number n such that G can be embedded in S_n. I'm curious to find >out what this number is. > >It can be shown with some counting arguments that n>=32 (I'm >happy to write these down but it's nicer if you thought about >this first). I would be surprised if n=32 but you never know. >-- >Michiel Boland >University of Nijmegen >The Netherlands My permutation group memory is rusty. Is it the "degree" of G you're asking for? I also get n = 32 as the smallest |S_n| divisible by |G|. I suspect one can argue |G| must divide |A_n| (still requiring only n >= 32), and if we can argue |G|*12 must divide |S_n| (accounting for all parities of edges and corners) we get n >= 33. On the flip side, I'll assert G has degree <= 42 (which is less than the obvious representation in S_48). If anyone can prove me either right or wrong, please do so. My assertion is based on the following hand-waving argument: G is a wreath product (or some kind of product) of the following subgroups: A: S_8 8! corner positions B: 3^7 3^7 corner orientations C: A_12 12!/2 edge positions D: 2^11 2^11 edge orientation Clearly subgroup A has degree 8 and C has degree 12. I claim (wave hands wildly:-) that BxD has degree at most 22, since it can be embedded in an S_22. I use every even factor of 22! for a component of 2^11, and every third factor of 22! for a component of 3^7. Divisibility arguments suggest some smaller S_n might embed 2^11x3^7, but I don't know whether such embeddings can be realized. This is an obvious area to consider for lowering the upper bound on degree(G). By divisibility we can conceivably embed 2^11x3^7 in an S_18, but not in an S_17. Finally, the degree of a wreath(?) product is bounded by the sum of the degrees of the multiplicands, hence degree(G) <= 8 + 12 + 22 = 42. If this (alleged) degree 42 construction is valid, is 42 minimal? I strongly doubt it.