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Mathematical Notes

, Volume 95, Issue 3–4, pp 443–449 | Cite as

The Hopfian property of n-periodic products of groups

  • S. I. AdianEmail author
  • V. S. Atabekyan
Article

Abstract

LetH be a subgroup of a groupG. A normal subgroupN H ofH 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 iI 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 iI 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 

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Yerevan State UniversityYerevanArmenia

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