Morphic p-groups

作者：Aliniaeifard Farid; Li Yuanlin^*; Nicholson W K

来源：Journal of Pure and Applied Algebra, 2013, 217(10): 1864-1869.

DOI：10.1016/j.jpaa.2013.01.014

A group G is called morphic if every endomorphism alpha : G -%26gt; G for which G alpha is normal in G satisfies G/G alpha congruent to ker(alpha). 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 p(n) (n %26gt; 3) to be abelian are given. Our main results show that if G is a morphic p-group of order p(n) with n %26gt; 3 such that either d(G) = 2 or vertical bar G%26apos;vertical bar %26lt; p(3), 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 p(4), p(5) and p(6) are abelian.

• 出版日期2013-10