On One Open Question In The Theory Of ?-subnormal And ?-abnormal Subgroups

All groups considered in this dissertation are finite.Let G be a group.A subgroup A of G is said to be ?-subnormal in G if there is a subgroup chain A=A0?A1?…?At=G,such that either Ai-1(?)Ai or Ai/(Ai-1)Ai is ?-primary for all i=1,...,t.A subgroup A of G is said to be ?-abnormal in G if L/KL is not ?-primary when-ever A?K<L?G.In this paper,we answer to Question 7.7 in and obtain the following result.Theorem 3.1 Every subgroup of G is either ?-subnormal or ?-abnormal in G if and only if G is a group of one of the following two types:(?)G is ?-nilpotent;(?)G=D(?)P,where:(a)D=G(?)?=G' is a ?-nilpotent ?-Hall subgroup of G;(b)P=NG(P)is a cyclic Sylow subgroup of G;(c)Z(G)is the unique maximal subgroup of P.