S-regular Group And Its Nature

The relationship between the subgroups of a finite group G and the group G itself have been extensively studied in the literature. In 1996, WANG [1] introduced the concept of c-normal subgroup which defined that a subgroup H is said to be a group G if there exist a normal subgroup K of G such that G =HK. and H K is contained in HG,where HG, is the maximal normal subgroup of G and used c -normality of maximal subgroups to determine the structures of some finite groups. ZHANG Xingjian notes that subnormal conditions are weaker than that of normality and uses subnormal condition to replace the normal conditions .It follows that s-normality of a finite group is defined in "On s-normal subgroups of finite groups" (To appear )and many new results are obtained.In this paper ,the author goes on with the research of [1] and determines the structures of some groups in which some primary subgroups is s-normal. All groups in this paper are finite .For notations and terminologies not given in this paper, the reader is referred to the texts of [2,3].In l,the author gives main definitions and basic results what are needed in the paper.In 2,the author determines the structures of some groups by using s-normality of Sylow subgroups and the maximal subgroups of Sylow subgroups .there are the main theorems that 1) A group G ismetanilpotent if and only if every Sylow subgroup of G is s normal in G . 2) Let H be a subgroup of a group. If every Sylow subgroup of H is s-normal in G .and G : H = 2 p ,where p is a prime and arepositive integers, then G is soluble. 3) Let H be a finite group and let P be a Sylow p-subgroup of G, where p is a prime divisor of the order of G such that ( G . p-1 ) = 1. Supper that there exists a maximal subgroup P1 of P such that P1 is s-normal in G and Op(G) P1,then G Op(G) is p-nilpotent.