Morphic p-groups

A group G is called morphic if every endomorphism α:G→G for which Gα is normal in G satisfies G/Gα≅ker(α). This concept for modules was first investigated by G. Ehrlich in 1976. Since then the concept has been extensively studied in module and ring theory. A recent paper of Li, Nicholson and Zan investigated the idea in the category of groups. A characterization for a finite nilpotent group to be morphic was obtained, and some results about when a small p-group is morphic were given. In this paper, we continue the investigation of the general finite morphic p-groups. Necessary and sufficient conditions for a morphic p-group of order pn(n>3) to be abelian are given. Our main results show that if G is a morphic p-group of order pn with n>3 such that either d(G)=2 or ∣G′∣