  Suppose $G$ is an arbitrary additively written primary abelian group with a
fixed large subgroup $L$. It is shown that $G$ is (a) summable; (b)
$\sigma$-summable;\break (c) a $\Sigma$-group; (d) $p^{\omega+1}$-projective
only when so is $L$. These claims extend results of such a kind obtained by
Benabdallah, Eisenstadt, Irwin and Poluianov, {\it Acta Math. Acad. Sci.
Hungaricae} (1970) and Khan, {\it Proc. Indian Acad. Sci. Sect. A} (1978).
