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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 H ∈ M is said to be absolutely closed in M if domMG(H) = H for every group G ∈ M 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−1 ∈ M, and Zpk ∉ M, where Zpk−1 , Zpk are cyclic groups of orders p(k−1), pk respectively. It is proved that a group H ∈ M 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 p ∈ M the statement ypk−1 ∈ Hpkr 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.
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