15 - The Thompson Chain of Subgroups of Co1

Summary

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 t_rs, an element of order 7 in some group where tsr=trs−1. We thus obtain a progenitor of shape

7⋆(n2):Sn

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 S_n, 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 S_n by the alternating group A_n, 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.