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This article is cited in 1 scientific paper (total in 1 paper)
BRIEF MESSAGE
Abelian groups with finite primary quotients
A. A. Fomin, A. V. Tsarev Moscow State Pedagogical University (Moscow)
Abstract:
An abelian group $A$ is called $\pi$-bounded for a set of prime numbers $\pi$, if all $p$-primary components $t_{p}(A/B)$ are finite for every subgroup $B\subset A$ and for every $p\in\pi$. E. V. Sokolov has introduced the class of $\pi$-bounded groups investigating $\mathcal{F}_{\pi}$-separable and $\pi^\prime$-isolated subgroups in the general group theory. The description of torsion $\pi$-bounded groups is trivial. E. V. Sokolov has proved that the description of mixed $\pi$-bounded groups can be reduced to the case of torsion free groups.
We consider the class of $\pi$-bounded torsion free groups in the present paper and we prove that this class of groups coincides with the class of $\pi$-local torsion free abelian groups of finite rank. We consider also abelian groups satisfying the condition $(\ast)$, that is such groups that their quotient groups don't contain subgroups of the form $\mathbb{Z}_{p^{\infty}}$ for all prime numbers $p\in\pi$, where $\pi$ is a fixed set of prime numbers. It is clear that all $\pi$-bounded groups satisfy the condition $(\ast)$. We prove that an abelian group $A$ satisfies the condition $(\ast)$ if and only if both groups $t(A)$ and $A/t(A)$ satisfy the condition $(\ast)$. We construct also an example of a non-splitting mixed group of rank $1$, satisfying the condition $(\ast)$, for every infinite set $\pi$ of prime numbers.
Keywords:
abelian group, separability of subgroups, $\pi$-bounded abelian group, $\pi$-local torsion free abelian group.
Received: 12.08.2021 Accepted: 21.12.2021
Citation:
A. A. Fomin, A. V. Tsarev, “Abelian groups with finite primary quotients”, Chebyshevskii Sb., 22:5 (2021), 400–406
Linking options:
https://www.mathnet.ru/eng/cheb1146 https://www.mathnet.ru/eng/cheb/v22/i5/p400
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