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

Abstract

AbstractLetGbe a finitep-group. We denote by$${{\,\mathrm{\mathcal {X}}\,}}_i(G)$$Xi(G)the intersection of all subgroups ofGhaving index$$p^i$$piinG. In this paper, the newly introduced series$$\{{{\,\mathrm{\mathcal {X}}\,}}_i(G)\}_i$${Xi(G)}iis investigated and a number of results concerning its behaviour are proved. As an application of these results, we show that if an abelian subgroupAofGintersects each one of the subgroups$${{\,\mathrm{\mathcal {X}}\,}}_i(G)$$Xi(G)at$${{\,\mathrm{\mathcal {X}}\,}}_i(A)$$Xi(A), thenAhas a complement inG. Conversely if an arbitrary subgroupHofGhas a normal complement, then$${{\,\mathrm{\mathcal {X}}\,}}_i(H) = {{\,\mathrm{\mathcal {X}}\,}}_i(G) \cap H$$Xi(H)=Xi(G)∩H.