Influences Of Subgroups With Conjugate Permutable Conditions And Minimal Or Maximal Subgroups Of Sylow Subgroups To The Structure Of A Group

Our investigation focuses on the influence of subgroups with conjugate permutable conditions and minimal or maximal subgroups of Sylow subgroups to the structure of groups, such as solvability, nilpotency and super-solvability.In "Conjugate permutable subgroups" (J. Algebra, 191(1997), 235-239) and "Groups with all cyclic subgroups are conjugate permutable subgroups" (J.Group Theory 2(1999), 47-51), T.Foguel introduced the following conceptDefinition 1 Let H be a subgroup of group G. If for any element g ∈ G, HH9= H9H holds. Then H is called a conjugate permutable subgroup of G, denoted byH <^C-PG.Some elementary results are obtained in the upper two papers. Recently, Mingyao Xu and Qinhai Zhang make some further study of conjugate permutable subgroups, especially ECP-groups, in "On conjugate permutable subgroups of a finite group" (to appear).The first part of his paper continues the study of the relation between groups and their conjugate permutable subgroups. Theorems of T.Foguel, Mingyao Xu and Qinhai Zhang, showing conjugate permutable subgroups of certain groups to be subnormal, are generalized to conjugate permutable subgroups either of a group satisfying the maximal and minimal condition on subgroups, or with finite index in any group.What is more, we introduce the following concept of itself conjugate permutable subgroup for the first time.Definition 2 Let H be a subgroup of G. If HH9 = H9H implies g ∈ N_G(H), then we call H an itself conjugate permutable subgroup of G, or H to be itself conjugate permutable in G.Some theorems for describing finite p-nilpotent or super-solvable groups are obtained by using this concept. Moreover, we define a new class of finite groups, named C2-groups, with the same concept. The structure of C2-groups is provided as well.Theorem 1 For a C2-group G, the following statements hold:(1) G is super-solvable;(2) Each Sylow subgroup of G with odd order is abelian, and Sylow 2-subgroup of G is either abelian or a hamiltonian group;(3) G' is abelian, hence G is meta-abelian.In the following part, we study further the method of localization for minimal subgroups, which localizes the property of minimal subgroups to the normalizer of any Sylow subgroup containing them. This method is introduced by Shirong Li in "On minimal subgroups of finite groups" (Com. in Algebra, 22(6) (1994), 1913-1918). By exchanging the condition "p is the smallest prime divisor of the order of G" with "p — 1 is prime to the order of G", a generalized result of Shirong's main result in "On minimal subgroups of finite groups(III)" (Com. in Algebra, 268(8) (1998), 2453-2461) is obtained.To show that the above result is a generalization with substance, an example is provided. In the end, we apply the method of localization for minimal subgroups to the maximal subgroups of Sylow subgroups and obtain a description for super-solvable groups.