##
On countable extensions of primary abelian groups

##
*
Peter Danchev
*

** Address.**
13, General Kutuzov Street, bl. 7, floor 2, apart. 4, 4003 Plovdiv, Bulgaria

** E-mail:**
pvdanchev@yahoo.com

pvdanchev@mail.bg

**Abstract.**
It is proved that if $A$ is an abelian $p$-group with a pure subgroup $G$ so that $A/G$ is at most countable and $G$ is either $p^{\omega+n}$-totally projective or $p^{\omega+n}$-summable, then $A$ is either $p^{\omega+n}$-totally projective or $p^{\omega+n}$-summable as well. Moreover, if in addition $G$ is nice in $A$, then $G$ being either strongly $p^{\omega+n}$-totally projective or strongly $p^{\omega+n}$-summable implies that so is $A$. This generalizes a classical result of Wallace (J. Algebra, 1971) for totally projective $p$-groups as well as continues our recent investigations in (Arch. Math. (Brno), 2005 and 2006).
Some other related results are also established.

**AMSclassification. **20K10, 20K15.

**Keywords. **Countable quotient groups, $\omega$-elongations, $p^{\omega+n}$-totally projective groups, $p^{\omega+n}$-summable groups.