We introduce a new class of Abelian groups which lies strictly between the classes of co-Hopfian groups and Dedekind-finite groups, calling these groups mono Dedekind-finite. We prove the surprising fact that in the torsion case the mono Dedekind-finite property coincides with the co-Hopficity, thus extending a recent result by Chekhlov–Danchev–Keef Sib. Math. J. (2026), to appear, and we construct a torsion-free mono Dedekind-finite group which is not co-Hopfian as well as a Dedekind-finite group which is not mono Dedekind-finite. Some other closely relevant things are also established. For example, we extend the construction due to Arnold–Rangaswamy Boll. Unione Mat. Ital. Sez. B, Artic. Ric. Mat. (8) 10 (3), 605–611 (2007) of a countable Butler group that is not completely decomposable to find a Butler group of countably infinite rank which is mono Dedekind-finite, but not completely decomposable.
Danchev et al. (Mon,) studied this question.