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

Abstract

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.