The influence of the totally permutability of subgroups on σ-nilpotent residual(子群的完全置换性对σ-幂零根的影响)

Abstract

Similar to the influence on nilpotent residual of the totally permutability of subgroups,naturally,we can study their influence on σ-nilpotent residual when nilpotent groups are generalized to σ-nilpotent groups.The intersection of all normal subgroups N of G such that G/N is σ-nilpotent is called σ-nilpotent residual of G,and is denoted as G𝒩σ. Let G=AB, where A and B are totally permutable subgroups of G,this paper gives some new conclusions that B normalises A𝒩σ and centralises A𝒩σ by using the concepts and theories of σ-nilpotent subgroups and σ-supersoluble subgroups,and by applying some properties and methods of complete Hall σ-set and finite group theory.(先将幂零群推广为σ-幂零群，再研究子群的完全置换性对σ-幂零上根的影响。群G的所有使G/N为σ-幂零群的正规子群N的交称为G的σ-幂零上根，记为G𝒩σ。设G=AB，其中A与B是完全置换的，利用子群的完全置换性质、σ-超可解群与σ-幂零群的概念和相关理论、完备Hall σ-集的性质以及有限群论的一些基本方法，给出了B正规化A的σ幂零根和中心化A的σ-幂零根的一些新的结论。)

Keywords

• σ-supersoluble groups(σ-超可解群)
• σ-nilpotent groups(σ-幂零群)
• totally permutablity(完全置换性)
• sylow subgroups(sylow子群)