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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.