On S-Normal Subgroups Of Finite Groups

It is well known that the relationship between the properties of G and the structure of G has been studied extensively. In [1], Wang introduced the concept of c-normal subgroup and used c-normality of maximal subgroups to determine the structures of some groups. In [10,13], the authors gave some conditions on which a group is nilpotent(solvable, supsolvable) by c- normality of subgroups.We know that the concept of c-normal subgroup is related to the concept of normal subgroup. Note that the subnormal conditions are weaker than that of normality, we try to use the subnormal conditions replace the normal conditions. It follows that s-normality of a group is defined. Using s-normality of maximal subgroups, Sylow subgroups and the maximal subgroups of Sylow subgroups, we determine the structures of some groups. Therefore many known results are generalized.All groups in this paper are finite. The terminology and notations employed agree with stand usage, as in [1,2,11]. In § 1, we give main definitions and basic results that are needed in the paper. In § 2, we determine the structures of somegroups by using s-normality of Sylow subgroups and the maximal subgroups of Sylow subgroups. There are the main theorems that 1) Let G be a finite group and P be a Sylow p-subgroup of G where p is a prime divisor of |G| . Suppose that there exists a maximal subgroup P1 of P such that P1 is s-normal in G. If (|G|, p-1)=1, then G has a Hall p'-subgroup. 2) Let G be a finite group and P be a Sylow 2-subgroup of G. Suppose that there exists a maximal subgroup P1 of P such that P1 is s-normal in G. Then G is solvable; 3) Let G be a finite group. If there exists a Sylow 2-subgroup P of G such that P is s-normal in G, then G is solvable.In § 3, we determine the structures of some groups by using s-normality of maximal subgroups. There are the main theorems that l)Let N be a nontrivial normal subgroup of a group G. Then N is solvable if and only if every maximal subgroup of G not containing N is s-normal in G; 2) Let G be a finite group. Then G is solvable if and only if there exists a solvable s-normal maximal subgroup M of G; 3) Let G be a finite group and Mo=1 where M is a maximal subgroup of G. Then G is solvable if and only if M is a supersolvable s-normal subgroup of G and |G:M|=r where r is a prime.