New Characterizations of Some Classes of Finite Groups

Let G be a finite group and 3 a formation of finite groups. We say that a subgroup H of G is J(h)-normal in G if there exists a normal subgroup T of G such that TIT is a normal Hall subgroup of G and (H boolean AND T)H-G/H-G is contained in the J-hypercenter Z(infinity)J(G/H-G) of G/HG. In this paper, we obtain some results about the J(h)-normal subgroups and use them to study the structure of finite groups. Some new characterizations of supersoluble groups, soluble groups and p-nilpotent groups are obtained and some known results are generalized.