English Russian
Известия
Izvestiya of Altai State
University Journal
The News of Altai State University

 Архив журнала «Известия АлтГУ», начиная с 2017 г., размещен на новой версии сайта http://izvestiya.asu.ru
 Актуальная информация о журнале размещена на новой версии сайта http://izvestiya.asu.ru

Print ISSN 1561-9443
On-line ISSN 1561-9451
Issues list
Table of Contents
Physical science
Mathematics and mechanics
About the Journal
Editorial board and Editorial advisory board
Regulations on reviewing research papers
Rules of the articles representation
Publication Ethics of the journal «Izvestiya of Altai State University»
 
1(89)2016
  MATHEMATICS AND MECHANICS

S.A. Shakhova

Absolutely Closed Groups in Quasivarieties of Abelian Groups

For an arbitrary quasivariety M of groups, a group G in M and a subgroup H of G we introduce a set domMG (H), which is referred to as the dominion of the subgroup H of the group G in the quasivariety M. Similarly for the set of all elements of G, each of them has equal images under any pair of homomorphisms from G into an arbitrary group M ∈ M which coincide on H. A group HM is said to be absolutely closed in M if domMG(H) = H for every group GM containing H as a subgroup. In this paper, the absolutely closed groups in the quasivarieties of Abelian groups are studied. Let M be an arbitrary quasivariety of Abelian groups, ξ(M) be a set of prime numbers p, for which of them there exists a natural number k = k(p) that the following conditions are true: Zpk−1M, and Zpk M, where Zpk−1 , Zpk are cyclic groups of orders p(k−1), pk respectively. It is proved that a group HM is absolutely closed in M if and only if the following is true: for any element y of infinite order, belonging to reduced subgroup Hr of the group H, and for any number pM the statement ypk−1Hpkr is true.

DOI 10.14258/izvasu(2016)1-34

Key words: quasivariety, Abelian group, dominion, absolutely closed group

Full text at PDF, 584Kb. Language: Russian.

SHAKHOVA S.A.
Altai State University (Barnaul, Russia)
E-mail: sashakhova@gmail.com

 

Print Edition of "Izvestiya of Altai State University" © 1996-2017 Altai State University.
All rights reserved. Any of parts of a journal or edition as a whole cannot be reprinted without the written sanction of the authors or publisher. On purchase of a journal to address to ASU publishing house:
Altai State University. 656049, 66 Dimitrova street, Barnaul, Russia. Telephone + 7 (3852) 366351.