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

S. I. Adian ; V. S. Atabekyan

Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 483-491

Résumé

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.

Détail

Keywords: Hopfian group, $n$-periodic product, periodic group, inheritably normal subgroup.