| Group extension theory arises from many problems, and has many applications. For example, the first step to study finite groups is to classify all finite simple groups. Then the second step is to study finite non-simple group, and the basic tool is the group extension theory. Also, in topology ones computes homology groups of complexes, for example, one uses M-V exact sequences to compute homology groups, then the group extension problem naturally arises. It is widely used in topology. From pure algebraic point of view, group extension theory has a long history since the 19th century. The theory is relatively mature, but not satisfactory. The classical theory focuses on the easy cases like cyclic extension, central extension, Abelian extension and etc. For the general cases, there is no good theory. Even for the simple case like Abelian extension, when one is given groups G and H, the problem of classifying all extensions of G through H is not easy using the classical theory.In this paper, we focus on Abelian extension problem. Since all G,H,K are abelian groups, this is an central extension. G is a semidirect product of K and H. This is one to one correspondent to homomorphisms from H to Aut (K). We discuss two cases:1. K and H are Zpn,Zpm, where p is a prime number,2. K and H are Z, Zn. And we get the complete classification of such G. |
