Conjugate Permutability And The Stucture Of Finite Groups

In the present article, based on conjugate permutability, semi-normality and C-normality of some subgroups (like Sylow subgroups, maximal subgroups and maximal subgroups of Sylow subgroups), we study the structure of G. We investigate three problems, the main contents are as follows:First, we study how the R-conjugate permutability of Sylow sugroups of factor sub-groups effects the structure of the finite groups. That is to say, let G be a finite group. A, B and R are such subgroups of G that G= AB. Based on the concept of R-conjugate-permutable, we investigated the relationship between the nilpotence of G and the R-conjugate-permutability of the Sylow subgroups of A and B.Second, we discuss the relationship between the structure of G and the conjugate per-mutability and semi-normality of the maximal subgroups(2-maximal subgroups). Namely, based on the concepts of conjugate-permutable subgroups and semi-normal subgroups this paper investigates the supersolvability of a group G under the condition that max-imal subgroups(2-maximal subgroups) of a group G are either conjugate-permutable or semi-normal.Third, combining with the Sylow subgroups and the maximal subgroups, we discuss the relationship between the structure of G and the conjugate permutability of maxi-mal subgroups of Sylow subgroups. That is to say, based on the concepts of conjugate-permutable subgroups and semi-normal subgroups (C-normal subgroups), this paper inves- tigates the supersolvability of a group G under the condition that every maximal subgroups of Sylow subgroups of a group G are either conjugate-permutable or semi-normal(C-normal subgroups).