Conway’s “Murder Weapon”

The following quote is an excerpt from an interview with John Conway:

Coxeter came to Cambridge and he gave a lecture, then
he had this problem for which he gave proofs for selected
examples, and he asked for a unified proof. I left the lecture
room thinking. As I was walking through Cambridge,
suddenly the idea hit me, but it hit me while I was in the middle of the road. When the idea hit me I stopped and a
large truck ran into me and bruised me considerably, and
the man considerably swore at me. So I pretended that
Coxeter had calculated the difficulty of this problem so precisely
that he knew that I would get the solution just in the
middle of the road. In fact. I limped back after the accident
to the meeting. Coxeter was still there, and I said, “You
nearly killed me.” Then I told him the solution. It eventually
became a joint paper. Ever since, I’ve called that theorem
“the murder weapon.” One consequence of it is that in a group if a^2 = b^3 = c^5 =(abc)^{-1}, then c ^{610} =1.

Does anyone know where one can find a statement of this theorem or the publication information on the paper Conway and Coxeter wrote?

This quote is from Math. Intelligencer 23 (2001), no. 2,pp.8-9.

Answer

According to Siobhan Roberts (2006) King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry p.359 (Google Books):

This is a simple statement of Coxeter’s “Murder Weapon”: If “A^p=B^q=C^r=ABC=1” defines a finite group, then “A^p=B^q=C^r=ABC=Z” implies Z^2=1. Conway; and Conway, interview, November 26, 2005; Conway with Coxeter and G. C. Shepherd, “The Centre of a Finitely Generated Group,” in Tensor, 1972 (contains the proof for “The Murder Weapon”).

The latter reference apparently means this citation: http://www.ams.org/mathscinet-getitem?mr=333001

MR333001 20F05

Conway, J. H.; Coxeter, H. S. M.; Shephard, G. C. The centre of a finitely generated group. Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. II. Tensor (N.S.) 25 (1972), 405–418; erratum, ibid. (N.S.) 26 (1972), 477.

Review: https://zbmath.org/0236.20029, written by Coxeter:

The polyhedral group (l,m,n), defined by a^l=b^m=c^n=abc=1, is finite if and only if g>0, where g=2(l^{-1}+m^{-1}+n^{-1}-1)^{-1}, and then its order is g. The same condition for finiteness applies also to the binary polyhedral group \langle l,m,n\rangle, defined by a^l=b^m=c^n=abc, whose order is 2g. In fact, the central element z=a^l=b^m=c^n=abc is of period 2. This last statement is not at all obvious. The first step is to observe that the Cayley diagram for (l,m,n) is identical with the coset diagram for \langle l,m,n\rangle relative to the subgroup generated by z. This diagram covers the inversive plane, or the 2-sphere, with a “map” which consists of g/l l-gons representing a^l, g/m m-gons representing b^m, g/n n-gons representing c^n, and g negatively oriented triangles representing abc. The procedure is to compare the results of shrinking the peripheral circuit (or any other circuit) to a single point (or “point-circuit”) in two distinct ways, corresponding to the two discs into which the circuit decomposes the 2-sphere. More symmetrically, any point-circuit (such as the point at infinity of the inversive plane) may be continuously deformed into another point-circuit by contracting over all the oriented polygons in the Cayley diagram. The same procedure is applied in more complicated cases. For instance, in the group for which z is central while a^l=z^p, b^m=z^q, c^n=z^r, abc=z^s and g (as defined above) is a positive integer, the period of z is g|pl^{-1}+qm^{-1}+rn^{-1}-s|.

Attribution
Source : Link , Question Author : Oiler , Answer Author : Community

Leave a Comment