The influence of the totally permutability of subgroups on σ-nilpotent residual

• Published in: Zhejiang da xue xue bao. Journal of Zhejiang University. Sciences edition. Li xue ban Vol. 51; no. 1; pp. 1 - 4
• Main Authors: Shi, Zhijie, Mao, Yuemei, Ma, Xiaojian
• Format: Journal Article
• Language: Chinese
• Published: Hangzhou Zhejiang University 01.01.2024; Zhejiang University Press
• Subjects: Group theory; Subgroups; sylow subgroups(sylow子群); totally permutablity(完全置换性); σ-nilpotent groups(σ-幂零群); σ-supersoluble groups(σ-超可解群)
• ISSN: 1008-9497
• DOI: 10.3785/j.issn.1008-9497.2024.01.001

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.