Absolutely Closed Groups in Quasivarieties of Abelian Groups

Abstract

For an arbitrary quasivariety M of groups, a group G in M and a subgroup H of G we introduce a set dom^M_G (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 dom^M_G(H) = H for every group G ∈ Mcontaining 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: Z_p^k−1 ∈ M, and Z_p^k ∉ M, where Z_p^k−1 , Z_p^k are cyclic groups of orders p^(k−1), p^k 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 subgroupH_r of the group H, and for any number p ∈ M the statement y^pk−1 ∈ H^pk_r is true.

DOI 10.14258/izvasu(2016)1-34