The $${{\,\mathrm{\mathcal {X}}\,}}$$ -series of a p-group and Complements of Abelian Subgroups

Let G be a finite p-group. We denote by $${{\,\mathrm{\mathcal {X}}\,}}_i(G)$$ the intersection of all subgroups of G having index $$p^i$$ in G. In this paper, the newly introduced series $$\{{{\,\mathrm{\mathcal {X}}\,}}_i(G)\}_i$$ 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 $${{\,\mathrm{\mathcal {X}}\,}}_i(G)$$ at $${{\,\mathrm{\mathcal {X}}\,}}_i(A)$$ , then A has a complement in G. Conversely if an arbitrary subgroup H of G has a normal complement, then $${{\,\mathrm{\mathcal {X}}\,}}_i(H) = {{\,\mathrm{\mathcal {X}}\,}}_i(G) \cap H$$ .