Y-groups via Transitive Extension

Abstract

Motivated by an earlier observation by J. H. Fischer, around 1980 B. Conway conjectured that the Coxeter diagram Y555 on Fig. 1 together with a single additional (so-called “spider”) relation (ab1c1ab2c2ab3c3)10=1 form a presentation for the wreath product of the Monster group M and a group of order 2. This conjecture was proved by S. P. Norton and the author in 1990. The original proof was rather involved, relying on simple connectedness results for certain diagram geometries, on numerous data obtained by coset enumeration on a computer, and on some delicate calculations with subgroups coming from the 26-node theorem. In the present work we follow an inductive approach to the identification of Y-groups by considering larger Y-groups as transitive extensions of smaller ones. Along these lines we obtain an alternative identification proof for the Y-groups which is almost computer-free: we refer to only one result of (double) coset enumeration. Our approach provides a uniform understanding of the Y-groups, particularly of features such as centres and redundant generators.