In the final chapter all the ideas of the book come together to produce the chain of subgroups of the Conway simple group Co1 that was previously referred to as the Suzuki chain. Since this construction emphatically reveals that the chain includes Co1 itself, we prefer to call it the Thompson chain as it was John Thompson who first noted that, with one exception, the normalizers of the groups in the chain are maximal in Co1. In a complete graph on n vertices we let the directed edge from vertex r to vertex s correspond to trs, an element of order 7 in some group where tsr=trs−1. We thus obtain a progenitor of shape

in which the symmetric group permutes the vertices. Initially, we include an additional automorphism of the free product that simply squares each of the symmetric generators whilst commuting with the Sn, but we eventually discard it as it is not needed. We must now decide what a triangle 〈t12, t23, t31〉 generates, and we realize that the unitary group U3(3) has all the necessary properties. Factoring by a single relation that ensures that triangles generate copies of this unitary group, we find that a complete 4-graph generates the Hall–Janko group, a complete 5-graph generates the Lie group G2(4), a complete 6-graph generates the triple cover of the Suzuki simple group and a complete 7-graph generates Co1. For n > 7 the group collapses, but if we replace the symmetric group Sn by the alternating group An, then we may proceed as far as n = 9 when a 9-graph also generates Co1. In this configuration a 3-cycle on three vertices lies in the centre of the triple cover of the Suzuki group generated by the edges on the other 6 vertices. We conclude by using MOG techniques to embed this whole configuration into the 24-dimensional representation of the Conway group ·O acting on the Leech lattice, modulo of course the central element of order 2.