Classification Theorems

Abstract

We have already seen some classification theorems in the earlier chapters. Theorem 10.16 classifies primitive groups with cofinite Jordan sets. Theorem 11.12 classifies all infinite permutation groups which are highly homogeneous but not highly transitive. Theorem 12.9 classifies infinite primitive permutation groups which have proper primitive Jordan sets. We have also seen many different examples of Jordan groups in the last three chapters. In this chapter we tie together all the different classes of examples by a theorem (due to Adeleke & Macpherson) which extends Theorems 11.12 and 12.9 and classifies all infinite primitive Jordan groups G. Theorem 11.12 tells us that if G is highly homogeneous but not highly transitive then it preserves a relation of one of the four types described in Section 11.3. If G has primitive Jordan sets and is not highly homogeneous then by Theorem 12.9 it follows that G preserves a relation of one of the four types described in Chapter 12. The classification theorem of Adeleke & Macpherson tells us that if G is not one of the types mentioned above, and is not highly transitive, then it must either be an automorphism group of a Steiner k-system (cf. Sec. 11.2) or it must preserve limit structures of certain specified kinds.