ABSTRACT
An almost completely decomposable group X is a finite essential extension of a completely decomposable group A. Hence there is an integer e such that eX C A, and one obtains a simplified image of X in the finite group A /e A , namely eX /eA . Are there any essential properties of X that can be studied by looking at the fi nite group A = A /eA together with its subgroup eX = e X /e A l The answer is no and yes. The group A has much less structure than the completely decompos able A. It is easily possible that A is a free Z/eZ-module with many endomorphisms and automorphisms while A has a very rigid structure with endomorphisms and automorphisms severely restricted by the fully invariant type subgroups A(r). Considering A and its subgroup eX is not good enough. But the approximation of (A ,e X ) to (A, AT) is closer if A is furnished with the distinguished subgroups A(t) together with eX . By looking at A with these distinguished subgroups it is possible to decide whether A is regulating in X (Proposition 8.1.9) or the regulator of X (Theorem 8.1.10), one can decide whether two almost completely decompos able groups X and Y with common regulator are isomorphic (Theorem 8.1.10), and one can see that there is a natural weakening of isomorphism, called typeisomorphism (Definition 8.1.14). Classification up to type-isomorphism becomes the study of finite abelian groups with distinguished subgroups. Type-isomorphism can be compared with isomorphism and it will become clear why and to what ex tend type-isomorphism simplifies the classification problem. It remains to be seen in later chapters that type-isomorphism is fine enough to preserve important prop erties.
