From mschoene@math.rwth-aachen.de Wed Dec 7 16:31:35 1994 Return-Path: Received: from samson.math.rwth-aachen.de by life.ai.mit.edu (4.1/AI-4.10) for /com/archive/cube-lovers id AA05994; Wed, 7 Dec 94 16:31:35 EST Received: from hobbes.math.rwth-aachen.de by samson.math.rwth-aachen.de with smtp (Smail3.1.28.1 #11) id m0rFSHJ-000MPOC; Wed, 7 Dec 94 20:43 MET Received: by hobbes.math.rwth-aachen.de (Smail3.1.28.1 #19) id m0rFSHJ-0000PsC; Wed, 7 Dec 94 20:43 PST Message-Id: Date: Wed, 7 Dec 94 20:43 PST From: "Martin Schoenert" To: cube-lovers@life.ai.mit.edu Subject: A lemma that is *not* Burnside's Dan Hoey writes his e-mail message of 1994/11/04 In January of this year, Jerry Bryan and I wrote of counting the number of M-conjugacy classes of Rubik's cube. In the sense that (for instance) there is really only one position 1 QT from start, even though that QT may be applied in twelve different ways, this task amounts to counting the true number of positions of the cube. The earlier discussion centered on calculations involving computer analysis of large numbers of positions. However, a look in Paul B. Yale's book _Geometry and Symmetry_ gave me a clue: the Polya-Burnside theorem is a tool that allows us to perform this calculation by hand. The lemma used is indeed often referred to as ``Burnside's lemma'', or if used to count combinatorical objects as ``Polya-Burnside lemma''. However Peter M. Neumann in his paper ``A lemma that is not Burnside's'' (1979) pointed out that this lemma was known long befor Burnside rediscovered it. The first appearance seems to be Cauchy's work ``Memoire sur diverses proprietes remarquables des substitutions regulieres ou irregulieres, et des systemes de substitutiones conjugees (suite)'' of 1845. But it appears in a rather obscure form. It appears in more explicit form in Frobenius' paper ``"Uber die Congruenz nach einem aus zwei endlichen Gruppen gebildeten Doppelmodul'' in 1887. This is well befor Burnside's paper ``On some properties of groups of odd order'' in 1900. Polya refined it in his paper ``Kombinatorische Anzahlbestimmungen f"ur Gruppen, Graphen und chemische Verbindungen'' in 1937, for use in the part of combinatorics which deals with counting problems. Peter M. Neumann thus proposed to call this lemma ``Cauchy-Frobenius lemma''. But this was somehow not accepted, and it is now usually called ``A lemma that is not Burnside's''. Have a nice day. Martin. -- .- .-. - .. -. .-.. --- ...- . ... .- -. -. .. -.- .- Martin Sch"onert, Martin.Schoenert@Math.RWTH-Aachen.DE, +49 241 804551 Lehrstuhl D f"ur Mathematik, Templergraben 64, RWTH, 52056 Aachen, Germany