Ukr. Mat. Zh. - 2012. - 64, № 6. - pp. 766-771
We prove some properties of $n$-totally projective abelian $p$-groups. Under some additional conditions for the group structure, we obtain an equivalence between the notions of $n$-total projectivity and strong $n$-total projectivity. We also show that $n$-totally projective $A$-groups are isomorphic if they have isometric $p^n$-socles.