1 - Groups & Generality

Summary

In section 0, we shall introduce the various chain conditions which are implied by stability: the ascending and the descending chain conditions for intersections of uniformly definable families of subgroups (these intersections will then also be uniformly definable), and the ∣T∣⁺-chain condition for arbitrary Λ-definable subgroups. As we shall want these conditions to hold in saturated models as well, they are necessarily uniform. We shall also mention chain conditions which hold under stronger stability assumptions: the descending chain condition for definable subgroups in ω-stable groups, the descending chain condition for connected Λ-definable subgroups in the superstable case, and the ascending chain condition for connected definable subgroups in theories of finite rank.

However, chain conditions are important in their own right, not only in connection with stabilitYi once they are established, no further use of stability will be made in that chapter (with the exception of Theorem 1.1.13, which has been included in this section because of its proximity to the nilpotency results there). In fact, a more algebraic setting might consist of a group together with a family of sections comprising all centralizers and closed under taking normalizers, quotients, and – in the case of an abelian group – the subgroups of n-divisible elements and elements of order n for all n < ω; we would then require the uniform chain condition on intersections of uniform subfamilies.