On co-Hopfian groups

A group is called co-hopfian if it is not isomorphic with a proper subgroup. The aim of this paper is to obtain sufficient conditions for a group to be co-hopfian or non-co-hopfian. For example, it is shown that a reduced soluble minimax group which is abelian-by-nilpotent-by-finite, but not nilpotentby-finite, cannot be co-hopfian. This leads to the construction of many finitely generated soluble coherent groups which are not polycyclic. On the other hand, examples of co-hopfian polycyclic groups which are not nilpotent-by-finite are given. In addition it is shown that a soluble-by-finite group satisfying the minimal condition on normal subgroups is co-hopfian.