Morphic p-groups

• Published in: Journal of pure and applied algebra Vol. 217; no. 10; pp. 1864 - 1869
• Main Authors: Aliniaeifard, Farid, Li, Yuanlin, Nicholson, W.K.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.2013
• Subjects: Primary
• ISSN: 0022-4049; 1873-1376
• DOI: 10.1016/j.jpaa.2013.01.014

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′∣<p3, then G is abelian, where d(G) is the minimal number of generators of G. As consequences of our main results we show that any morphic p-groups of order p4,p5 and p6 are abelian.