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

TL;DR

This paper investigates the Hopfian property of $n$-periodic products of groups, establishing conditions under which these groups are Hopfian and constructing examples of nonresidually finite Hopfian groups with bounded exponents.

Contribution

It provides new criteria for inheritably normal subgroups in $n$-periodic products and proves that almost all such groups are Hopfian for odd $n \\ge 665$, enabling the construction of new group examples.

Findings

Normal subgroups contain $G^n$ if and only if they contain $G_i^n$

For odd $n \\ge 665$, all nontrivial normal subgroups contain $G^n$

Almost all $n$-periodic products are Hopfian

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.