The Influence Of Special Subgroups On The Structure Of Finite Groups

When we do research about the structure of a group, the influence of subgroups is very important. We can get a large number of information about the structure of a group from the properties of its subgroups. Thus, subgroups play an important part in the research on group theory. In particular, when we research into the structure of a finite group, we can get a lot of real information about the structure of a finite group by utilizing the properties of some special subgroups such as: Sylow subgroup, maximal subgroup of Sylow subgroup, maximal subgroup, 2-maximal subgroup,Hall—subgroup and so on. Many scholars of group theory at home and abroad have done some work about this aspect, such as [1],[4],[8,][9] of the references and so on.The purpose of the paper is to go further into the influence of special subgroups on the stucture of finite groups on the bases of [1] and [8]. This paper is composed of four chapters.Chapter 1 is the introduction of this paper, which introduces the progress of finite groups and the main work in this paper.In chapter 2, we investigate the structure of finite groups under the condition that all special subgroups are conjugate-permutable subgroups and obtain some sufficient conditions for solvability, Abelian property of finite groups. Some results modify and improve some results in [1] and [4]. Some results are new. we obtain some main results as follows:Theorem 2.3.1 Suppose that H is a π-Hall subgroup with even order ofG. If H and every Sylow subgroup of H are conjugate-permutable in G, then G is solvable.Theorem 2.3.2 Suppose that H is a 7r-Hall subgroup with even order of G. If there exists a maximal subgroup M of H is conjugate-permutable in G, and for any two elements x,y € H, only if (o(x),o(y))=l,[x,y]=l, then G is solvable.Theorem 2.3.3 Suppose that H is a ?r —Hall subgroup with even order of G. If any subgroup of Nq(H) is conjugate-permutable in G, then G is solvable.Theorem 2.3.4 If every maximal subgroup of G is conjugate-permutable in G, and is simple group, then G is Abelian group.Theorem 2.3.5 If there exists a solvable 2-maximal subgroup A ^ 1 of G which is conjugate-permutable in G, then G is solvable.Theorem 2.3.6 Suppose that P is a Sylow p—subgroup of G. If P is conjugate-permutable in G and the maxiaml subgroup of G/P is 1, then G is solvable.Theorem 2.3.7 If any maximal subgroup of every Sylow subgroup of G is conjugate-permutable in G, then G is solvable.Theorem 2.3.8 If any Sylow subgroup of every maximal subgroup of G is conjugate-permutable in G and at least there exists a maximal subgroup of G which is conjugate-permutable in G, then G is solvable.Theorem 2.3.9 If every 2-maximal subgroup of G is conjugate-permutable in G and at least there exists a maximal subgroup of G which is conjugate-permutable in G, then G is solvable.In chapter 3, we mainly discuss the influence of c-normality of special subgroups on the stucture of finite groups on the bases of [8] and [9] and obtain some sufficient conditions for solvability, supersolvability of finite groups, weobtain some main results as follows:Theorem 3.3.1 Suppose that R is a minimal subgroup of G such that G/R is supersolvable. If R\ < R and Ri is c—normal in G, then G is super-solvable.Theorem 3.3.2 Suppose that R is a minimal subgroup of G such that G/R is supersolvable. If there exists a minimal normal subgroup of G which is c—normal in G, then G is supersolvable.Theorem 3.3.3 Suppose that K is a tt — Hall subgroup with even order of G. If K is conjugate-permutable in G and any maximal subgroup M of G which does not contain K is c-normal in G, then G is solvable.In chapter 4, we mainly give some properties of Carter subgroup and some theorems, we obtain some main results as follows:Theorem 4.3.1 There does not exist real Carter subgroup in a finite nilpotent group.Theorem 4.3.2 Suppose that G is a finite group. If H is a Carter subgroup of G, then H is a maximal nilpotent subgroup of G.Theorem 4.3.3 Suppose that K < G an,d if is a tt — Hall subgroup of G. If any Sylow subgroup of G is conjugate-permutable in G and Nq{H) < K, then K is a Carter subgroup of G.