# The Hopfian property of *n*-periodic products of groups

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## 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 } ∩ *H*. It is proved in the paper that a subgroup \(N_{G_i }\) of a factor *G* _{ i } of the *n*-periodic product Π _{ i∈I } ^{ n } *G* _{ 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* ≥ 665 every nontrivial normal subgroup in a given *n*-periodic product *G* = Π _{ i∈I } ^{ n } *G* _{ i } contains the subgroup *G* ^{ n }. It follows that almost all *n*-periodic products *G* = *G* _{1} _{*} ^{ n } *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## Preview

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