Subgroups generated by images of endomorphisms of Abelian groups and duality

Abstract

Abstract A subgroup H of a group G is called endo-generated if it is generated by endo-images, i.e. images of endomorphisms of G. In this paper we determine the following classes of Abelian groups: (a) the endo-groups, i.e. the groups all of whose subgroups are endo-generated; (b) the endo-image simple groups, i.e. the groups such that no proper subgroup is an endo-image; (c) the pure-image simple, i.e. the groups such that no proper pure subgroup is an endo-image; (d) the groups all of whose endo-images are pure subgroups; (e) the ker-gen groups, i.e. the groups all of whose kernels are endo-generated. Some dual notions are also determined.