Abelian groups with finite primary quotients

Abstract

An abelian group 𝐴 is called 𝜋-bounded for a set of prime numbers 𝜋, if all 𝑝-primary components 𝑡𝑝(𝐴/𝐵) are finite for every subgroup 𝐵 ⊂ 𝐴 and for every 𝑝 ∈ 𝜋. E. V. Sokolov has
introduced the class of 𝜋-bounded groups investigating ℱ𝜋-separable and 𝜋′-isolated subgroups in the general group theory. The description of torsion 𝜋-bounded groups is trivial. E. V. Sokolov has proved that the description of mixed 𝜋-bounded groups can be reduced to the case of torsion free groups. We consider the class of 𝜋-bounded torsion free groups in the present paper and we prove that this class of groups coincides with the class of 𝜋-local torsion free abelian groups of finite rank.
We consider also abelian groups satisfying the condition (*), that is such groups that their quotient groups don’t contain subgroups of the form Z𝑝∞ for all prime numbers 𝑝 ∈ 𝜋, where 𝜋
is a fixed set of prime numbers. It is clear that all 𝜋-bounded groups satisfy the condition (*).
We prove that an abelian group 𝐴 satisfies the condition (*) if and only if both groups 𝑡(𝐴) and 𝐴/𝑡(𝐴) satisfy the condition (*). We construct also an example of a non-splitting mixed group
of rank 1, satisfying the condition (*), for every infinite set 𝜋 of prime numbers.