Finite morphic $p$-groups

TL;DR

This paper characterizes finite morphic p-groups, showing that the only such groups are homocyclic p-groups and a specific nonabelian group, extending previous results under weaker assumptions.

Contribution

It proves that the only finite morphic p-groups are homocyclic and a particular nonabelian group, broadening earlier classifications with less restrictive conditions.

Findings

Finite morphic p-groups are either homocyclic or a specific nonabelian group.

Previous classifications are extended under weaker hypotheses.

The result confirms the uniqueness of these groups as morphic p-groups.

Abstract

According to Li, Nicholson and Zan, a group $G$ is said to be morphic if, for every pair $N_{1}, N_{2}$ of normal subgroups, each of the conditions $G/N_{1} \cong N_{2}$ and $G/N_{2} \cong N_{1}$ implies the other. Finite, homocyclic $p$-groups are morphic, and so is the nonabelian group of order $p^{3}$ and exponent $p$, for $p$ an odd prime. It follows from results of An, Ding and Zhan on self dual groups that these are the only examples of finite, morphic $p$-groups. In this paper we obtain the same result under a weaker hypotesis.