Finite morphic $p$-groups

A. Caranti, C. M. Scoppola

Published 2014-11-04Version 1

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. In this paper we show that these are the only examples of finite, morphic $p$-groups.