On the existence of complements in a group to some abelian normal subgroups

A complement to a proper normal subgroup $H$ of a group $G$ is a subgroup $K$ such that $G=HK$ and $H\cap K=\langle 1\rangle$. Equivalently it is said that $G$ splits over $H$. In this paper we develop a theory that we call hierarchy of centralizers to obtain sufficient conditions for a group to split over a certain abelian subgroup. We apply these results to obtain an entire group-theoretical wide extension of an important result due to D. J. S. Robinson formerly shown by cohomological methods.