On The Almost almost Dense and Pure-4 Subgroups of Abelian Groups

Abstract

We gave the some general properties of anew subgroups which are called pure (1-2-3) subgroups of abelian group G. And he gave some new results of minimal neat subgroups of abelian groups. “L. Fuchs” poses the problem of characterizing the subgroups of an abealian group G which are intersections of finitely many pure subgroups of G (problem 13, p.134).This problem has been solved by “Khalid ben abdullah” and John Irwin.In this paper we shall give the generalization of the problem solved by Khalid Ben abdullah. Firstly we shall give the definition of new subgroups which are a some family of the pure subgroups, and we shall study the some general properties of this new type of the purity.