Formations of Finite Groups in Polynomial Time: the F-Hypercenter

For a wide family of formations F (which includes Baer-local formations) it is proved that the F-hypercenter of a permutation finite group can be computed in polynomial time. In particular, the algorithms for computing the F-hypercenter for the following classes of groups are suggested: hereditary local formations with the Shemetkov property, rank formations, formations of all quasinilpotent, Sylow tower, p-nilpotent, supersoluble, w-supersoluble and $SC$-groups. For some of these formations algorithms for the computation of the intersection of all maximal F-subgroups are suggested.