Finite groups with some HH-subgroups

A subgroup HH is said to be an HH-subgroup of a finite group GG if Hg∩NG(H)≤HHg∩NG(H)≤H for all g∈Gg∈G. For every prime pp dividing the order of GG, let PP be a Sylow pp-subgroup of GG and DD a subgroup of PP with 1<|D|<|P|1<|D|<|P|. We investigate the structure of GG under the assumption that each subgroup HH of PP with |H|=|D||H|=|D| is an HH-subgroup of GG. Some earlier results are generalized. Some results about formation are obtained.