On p-nilpotence and supersolvability of finite groups

A subgroup K of a finite group G is called an H-subgroup of G if the following condition is satisfied: N-G(K) boolean AND K-g <= K for all g is an element of G. The set of all H-subgroups of a finite group G will be denoted by H(G). In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to H(G).