The Properties Of Subgroups And The Structure Of Finite Groups

In recent years, the investigation to nilpotency, p-nilpotency, solvablity, super-solvablity of finite groups through various generalized normality is a very active topic. There are many famous results on this topic. For nilpotency, the famous result is:Let G be a finite group, then G is nilpotent if and only if 1)every maximal subgroup of G is normal in G or 2)every Sylow subgroup of G is normal in G. For p-nilpotency, Ito's famous result(see [2]) is:Let G be a finite group and p be a prime dividing the order of G, if every element of G with order p is in Z(G), and moreover when p=2, every element of G with order 4 is in Z(G), then G is p-nilpotent; Burnside's famous result is:Let G be a finite group and p be a prime dividing the order of G and P be a Sylow p-subgroup of G, if P is in Z(NG(P)), then G is p-nilpotent. Huppert's famous result for supersolvablity is:Let G be a finite group, then G is supersolvable if and only if the index of every maximal subgroup of G in G is a prime. In 1996, Yanming Wang (see [28])introduced the concept of c-normal subgroup of a finite group G and proved that a finite group G is solvable if and only if every maximal subgroup of G is c-normal in G. In 1998, Xiangying Su (see [1])introduced the concept of seminormal subgroup of a finite group G and proved that a finite group G is supersolvable if and only if every maximal subgroup of G is seminormal in G. Since then, a lot of investigations were made on seminormal subgroups of finite groups, many interesting results were obtained(see [1], [12-14], [21-27]).In 1997, Foguel (see [14])introduced the concept of conjugate subgroup of a finite group G. Since then, a lot of investigations were made on seminormal sub-groups of finite groups, many interesting results were obtained(see [16], [29-32]). In [33], the author introduced the concept of weakly c-normal subgroup of a finite group G, then in [34],the author made some investigations on weakly c-normal subgroups of finite groups and got some interesting results. Basing on the former work, the authors try to make some further researches on seminormal subgroups of finite groups, and obtained some sufficient conditions of finite groups to be nilpotent, p-nilpotent, supersolvable or solvable. The results are Theorems 1.1-1.16; Theorems 2.1-2.6 and Theorems 3.1-3.4.