From:

~~~ ~~~ Subject:

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