Conway's "Murder Weapon"

Solution 1:

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 $g|pl^{-1}+qm^{-1}+rn^{-1}-s|$.

Solution 2:

A Google search for "Conway and Coxeter murder" came up with the book "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry".

A search via Google inside this book came up with this on page 359, note 14:

This is a simple consequence 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 $z^2 = 1$."

The reference is Conway, Coxeter, and Shepherd, Tensor, 1972.