(untitled) (European Journal of Pure and Applied Mathematics, 2025)
In this paper, we introduce the superfluous kernel property in order to characterize generalized Hopfian groups. Then, we state our first theorems in this regard, through the study of two categories of Abelian groups, namely the reduced p-groups and the reduced torsion groups. In fact, we answer to the open question about the implication from generalized Hereditarily Hopficity to finiteness. Additionally, we succeed to prove a third theorem for the category of the divisible p-groups. Along all these results, we also try to benefit from the properties of Hereditarily Hopfian groups to easily reach the generalized hopficity property.