The Hopfian Property of $n$-Periodic Products of Groups

Abstract

Let $H$ be a subgroup of a group $G$. A normal subgroup $N_H$ of $H$ is said to be inheritably normal if there is a normal subgroup $N_G$ of $G$ such that $N_H=N_G\cap H$. It is proved in the paper that a subgroup $N_{G_i}$ of a factor $G_i$ of the $n$-periodic product $\prod_{i\in I}^nG_i$ with nontrivial factors $G_i$ is an inheritably normal subgroup if and only if $N_{G_i}$ contains the subgroup $G_i^n$. It is also proved that for odd $n\ge 665$ every nontrivial normal subgroup in a given $n$-periodic product $G=\prod_{i\in I}^nG_i$ contains the subgroup $G^n$. It follows that almost all $n$-periodic products $G=G_1\overset{n}{\ast}G_2$ are Hopfian, i.e., they are not isomorphic to any of their proper quotient groups. This allows one to construct nonsimple and not residually finite Hopfian groups of bounded exponents.