Recognition Theorems

Abstract

As we have described in Section 1.1, the final stage of every classification theorem involves an “identification” of the group G under investigation with some known simple group G*. Moreover, this identification is made by means of a set of intrinsic conditions which serve to “characterize” G* among simple K-groups. For example, the description of Conway’s groups in terms of the Leech lattice is extrinsic, since it involves an action of the groups on a geometry whose definition is given independently of the groups themselves. To obtain an intrinsic characterization of any of these groups, one must either show that it is possible to reconstruct the Leech lattice solely from information about their subgroups or else prove that their multiplication tables are uniquely determined by their subgroup structure.