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Mat. Zametki, 2014, Volume 95, Issue 4, Pages 483–491 (Mi mz10429)  

This article is cited in 3 scientific papers (total in 3 papers)

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

S. I. Adiana, V. S. Atabekyanb

a Steklov Mathematical Institute of the Russian Academy of Sciences
b Yerevan State University

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 $N_H=N_G\cap H$.
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.

Funding Agency Grant Number
State Committee on Science of the Ministry of Education and Science of the Republic of Armenia 13-RF-030
Russian Foundation for Basic Research


DOI: https://doi.org/10.4213/mzm10429

Full text: PDF file (491 kB)
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English version:
Mathematical Notes, 2014, 95:4, 443–449

Bibliographic databases:

Document Type: Article
UDC: 512.54
Received: 25.10.2013

Citation: S. I. Adian, V. S. Atabekyan, “The Hopfian Property of $n$-Periodic Products of Groups”, Mat. Zametki, 95:4 (2014), 483–491; Math. Notes, 95:4 (2014), 443–449

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71  mathnet  crossref  crossref  isi  elib
    2. S. I. Adian, Varuzhan Atabekyan, “Characteristic properties and uniform non-amenability of $n$-periodic products of groups”, Izv. Math., 79:6 (2015), 1097–1110  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    3. S. I. Adian, V. S. Atabekyan, “Periodic products of groups”, J. Contemp. Math. Anal.-Armen. Aca., 52:3 (2017), 111–117  crossref  mathscinet  zmath  isi  scopus
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