FI-extending hulls of Abelian groups

All groups considered are Abelian. As is well known, in a divisible group every subgroup is essential in a direct summand. Moreover, the fully invariant subgroups play a crucial role in the structure of an Abelian group. Thus it is natural to consider the class of groups in which every fully invariant subgroup is essential in a direct summand. In this paper, we provide a construction for imbedding a torsion group into a FI-extending group in some "minimal" way. This imbedding preserves p-height.