# 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.

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 “$$A^p=B^q=C^r=ABC=1A^p=B^q=C^r=ABC=1$$” defines a finite group, then “$$A^p=B^q=C^r=ABC=ZA^p=B^q=C^r=ABC=Z$$” implies $$Z^2=1Z^2=1$$. 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 $$(l,m,n)(l,m,n)$$, defined by $$a^l=b^m=c^n=abc=1a^l=b^m=c^n=abc=1$$, is finite if and only if $$g>0g>0$$, where $$g=2(l^{-1}+m^{-1}+n^{-1}-1)^{-1}g=2(l^{-1}+m^{-1}+n^{-1}-1)^{-1}$$, and then its order is $$gg$$. The same condition for finiteness applies also to the binary polyhedral group $$\langle l,m,n\rangle\langle l,m,n\rangle$$, defined by $$a^l=b^m=c^n=abca^l=b^m=c^n=abc$$, whose order is $$2g2g$$. In fact, the central element $$z=a^l=b^m=c^n=abcz=a^l=b^m=c^n=abc$$ is of period $$22$$. This last statement is not at all obvious. The first step is to observe that the Cayley diagram for $$(l,m,n)(l,m,n)$$ is identical with the coset diagram for $$\langle l,m,n\rangle\langle l,m,n\rangle$$ relative to the subgroup generated by $$zz$$. This diagram covers the inversive plane, or the $$22$$-sphere, with a “map” which consists of $$g/lg/l$$ $$ll$$-gons representing $$a^la^l$$, $$g/mg/m$$ $$mm$$-gons representing $$b^mb^m$$, $$g/ng/n$$ $$nn$$-gons representing $$c^nc^n$$, and $$gg$$ negatively oriented triangles representing $$abcabc$$. 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 $$22$$-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 $$zz$$ is central while $$a^l=z^pa^l=z^p$$, $$b^m=z^qb^m=z^q$$, $$c^n=z^rc^n=z^r$$, $$abc=z^sabc=z^s$$ and $$gg$$ (as defined above) is a positive integer, the period of $$zz$$ is $$g|pl^{-1}+qm^{-1}+rn^{-1}-s|g|pl^{-1}+qm^{-1}+rn^{-1}-s|$$.