ABSTRACT
In this chapter a weakening of isomorphism is introduced and studied. The concepts of “near-isomorphism” and “stable equivalence” were introduced by Lady ([Lad74], [Lad75]). It will be shown that near-isomorphism coincides with typeisomorphism for almost completely decomposable groups. We saw in Chapter 8 that type-isomorphism, and so near-isomorphism, is a weakening of isomorphism that substantially simplifies classification and does not require the exact knowledge of the divisibilities of types, an insurmountable problem in principle. We will further study the common features of nearly isomorphic groups and find that they share many important properties. Most prominently, near-isomorphic groups have very similar decomposition properties and the deepest results in the theory of almost completely decomposable groups were obtained by combining classification up to near-isomorphism with the preservation principles of decompositions.
