Projectively Invariant Subgroups of Abelian $p$-Groups

For a projectively invariant subgroup $C$ of a reduced $p$-group $G$, a nondecreasing sequence of ordinals and the symbol $\infty$ is constructed in which the $k$th position, $k=0,1,2,\dots$, is occupied by the minimum of heights in $G$ of all nonzero elements of the subgroup $p^kC[p]$. It is proved that if all elements of this sequence are integers, then the subgroup $C$ is fully invariant.