Countable extensions of torsion abelian groups

Peter Danchev

13, General Kutuzov Street, block 7, floor 2, ap. 4, 4003 Plovdiv, Bulgaria


Suppose $A$ is an abelian torsion group with a subgroup $G$ such that $A/G$ is countable that is, in other words, $A$ is a torsion countable abelian extension of $G$. A problem of some group-theoretic interest is that of whether $G \in \mathbb K$, a class of abelian groups, does imply that $A\in \mathbb K$. The aim of the present paper is to settle the question for certain kinds of groups, thus extending a classical result due to Wallace (J. Algebra, 1981) proved when $\mathbb K$ coincides with the class of all totally projective $p$-groups.

AMSclassification. 20K10, 20K20, 20K21.

Keywords. Countable factor-groups, $\Sigma$-groups, $\sigma$-summable groups, summable groups, $p^{\omega + n}$-projective groups.