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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 . 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 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.
We describe the method of symmetric generation of groups that has been used to produce pleasing, well-motivated constructions of many of the sporadic simple groups. A progenitor 2*n : N is defined to be a free product of n copies of the cyclic group of order 2, extended by a permutation group N of degree n that permutes these n involutory symmetric generators by conjugation. A useful lemma is introduced that leads to suitable relations by which to factor the infinite progenitor in order to produce a finite image that contains n distinct involutions permuted by an isomorphic copy of N. These ideas are employed using .
In the first case, we have and the symmetric generators correspond 4 to tetrads of points of Ω. Factoring this by a single short relation results in the Conway group ·O and the symmetric generators spontaneously reveal themselves to be the negatives of the Conway elements ξT.
In the second case, we have n = 3795 and the symmetric generators correspond to the trios. The Lemma leads us to two simple relations and factoring by them results in the largest Janko group J4. In fact one of the relations is essentially redundant and John Bray has shown that omitting it leads to a group J4 × 2. In order to prove these assertions in the two cases, we first use known representations to verify that the group claimed is indeed an image of the progenitor and satisfies the additional relations. We then use the double coset enumerator of John Bray and the author, written in Magma, to show that the image has the right order. Unfortunately in both these cases the group is too small to enumerate double cosets of form NwN and so we use theoretical arguments to identify subgroups
and
, respectively, and enumerate double cosets of form HwN.
