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 Mat. Zametki, 2017, Volume 101, Issue 3, Pages 323–329 (Mi mz10581)

On Residually Finite Groups of Finite General Rank

D. N. Azarov

Ivanovo State University

Abstract: Following A. I. Maltsev, we say that a group $G$ has finite general rank if there is a positive integer $r$ such that every finite set of elements of $G$ is contained in some $r$-generated subgroup. Several known theorems concerning finitely generated residually finite groups are generalized here to the case of residually finite groups of finite general rank. For example, it is proved that the families of all finite homomorphic images of a residually finite group of finite general rank and of the quotient of the group by a nonidentity normal subgroup are different. Special cases of this result are a similar result of Moldavanskii on finitely generated residually finite groups and the following assertion: every residually finite group of finite general rank is Hopfian. This assertion generalizes a similar Maltsev result on the Hopf property of every finitely generated residually finite group.

Keywords: group of finite rank, residual finiteness.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation

DOI: https://doi.org/10.4213/mzm10581

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Bibliographic databases:

UDC: 512.543
Revised: 10.06.2016

Citation: D. N. Azarov, “On Residually Finite Groups of Finite General Rank”, Mat. Zametki, 101:3 (2017), 323–329

Citation in format AMSBIB
\Bibitem{Aza17} \by D.~N.~Azarov \paper On Residually Finite Groups of Finite General Rank \jour Mat. Zametki \yr 2017 \vol 101 \issue 3 \pages 323--329 \mathnet{http://mi.mathnet.ru/mz10581} \crossref{https://doi.org/10.4213/mzm10581} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3635425} \elib{http://elibrary.ru/item.asp?id=28405142}