Geodesic
The Hopfian Property of $n$-Periodic Products of Groups
Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 483-491.

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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.
Keywords: Hopfian group, $n$-periodic product, periodic group, inheritably normal subgroup. 
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S. I. Adian; V. S. Atabekyan. The Hopfian Property of $n$-Periodic Products of Groups. Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 483-491. http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a0/

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