Let $H$ be a subgroup of a group $G$. A normal subgroup $N_H$ of $H$ is saidto 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 afactor $G_i$ of the $n$-periodic product $\prod_{i\in I}^nG_i$ with nontrivialfactors $G_i$ 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 $n\ge 665$ everynontrivial normal subgroup in a given $n$-periodic product $G=\prod_{i\inI}^nG_i$ contains the subgroup $G^n$. It follows that almost all $n$-periodicproducts $G=G_1\overset{n}{\ast}G_2$ are Hopfian, i.e., they are not isomorphicto any of their proper quotient groups. This allows one to construct nonsimpleand not residually finite Hopfian groups of bounded exponents.
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机译：假设$ H $是组$ G $的子组。如果存在$ G $的正常子组$ N_G $使得$ N_H = N_G \ cap H $，则$ H $的正常子组$ N_H $可以继承为正常。在本文中证明，具有非平凡因素$ G_i $的$ n $周期乘积$ \ prod_ {i \ in I} ^ nG_i $的因数$ G_i $的子组$ N_ {G_i} $是可继承的正常子组当且仅当$ N_ {G_i} $包含子组$ G_i ^ n $。还证明了对于给定的$ n $周期积$ G = \ prod_ {i \ inI} ^ nG_i $中奇数$ n \ ge 665 $的每个非平凡正常子组包含子组$ G ^ n $。因此，几乎所有$ n $-周期积$ G = G_1 \ overset {n} {\ ast} G_2 $都是Hopfian，即它们与它们的任何适当商组都不同构。这允许构造有界指数的非简单且不是残差有限的Hopfian组。
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