Some Characterizations about the Structure of Finite Groups.

A subgroup H of a finite group G is called an ℋ-subgroup of G if N_G(H) ∩ H^g ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G). [ABSTRACT FROM AUTHOR]