The influence of H-subgroups on the structure of finite groups

A subgroup H is said to be an H-subgroup of a group G if H(g) boolean AND N(G)(H) <= H for all g is an element of G. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order p(m) of a Sylow p-subgroup of G belongs to H(G) for a given positive integer m. Some results related to p-nilpotence and supersolvability of a group G are obtained.