The $\mathcal{X}$-series of a $p$-group and complements of abelian subgroups

Let $G$ be a $p$-group. We denote by ${\mathcal{X}}_i(G)$ the intersection of all subgroups of $G$ having index $p^i$, for $i\le {\log }_p(\mathrm{\mid }G\mathrm{\mid })$. In this paper, the newly introduced series $\{{\mathcal{X}}_i(G){\}}_i$ is investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroup $A$ of $G$ intersects each one of the subgroups ${\mathcal{X}}_i(G)$ at ${\mathcal{X}}_i(A)$, then $A$ has a complement in $G$. Conversely if an arbitrary subgroup $H$ of $G$ has a normal complement, then ${\mathcal{X}}_i(H)={\mathcal{X}}_i(G)\cap H$.