11 - The Conway Group ·O

Summary

The Conway group ·O is defined to be the group of all symmetries of Λ fixing the origin. The manner in which Λ has been constructed ensures that it is preserved by all permutations of M₂₄ and all sign changes on a C; we now introduce the Conway element ξ_T that is not in the aforementioned group of shape 2¹² : M₂₄ and which we can show preserves Λ. We use ξ_T to prove transitivity on type 2 vectors, type 3 vectors and crosses, and are then able to work out the order of ·O. This construction is now repeated using the algebra package Magma and a full explanation of the computation is included. An important Remark explains how every element of ·O corresponds in a certain sense to one of the crosses. The stabilizers of a type 2 vector and of a type 3 vector are the Conway groups Co₂ and Co₃, respectively. Each of these sporadic groups is described in some detail, both manually and computationally using Magma. The Classification of Finite Simple Groups (CFSG) states that any finite simple group is either a member of one of the known infinite families or it is one of 26 sporadic groups. We give a brief description of these groups and how they were discovered, and refer the reader to more complete expositions. We conclude by mentioning the remarkable observation made by John McKay that relates the modular function j to the degrees of irreducible representations of the Monster group M. Conway referred to this tantalizing connection with number theory as Monstrous Moonshine.