τ-拟置换子群对有限群结构的影响  

The Influence ofτ-quasipermutable Subgroups on the Structure of Finite Groups

作　　者：施智杰[1] 毛月梅[1] SHI Zhi-jie;MAO Yue-mei(School of Mathematical and Computer Science,University of Datong of Shanxi,Datong 037009,China)

机构地区：[1]山西大同大学数计学院,山西大同037009

出　　处：《数学的实践与认识》2020年第11期211-217,共7页Mathematics in Practice and Theory

基　　金：国家自然科学青年基金(11901364);山西大同大学博士科研启动基金(2016-B-04);山西大同大学2018年度科研基金(2018K10)。

摘　　要：群G的一个子群H称为τ-拟置换的,如果G有一个子群B满足G=NG(H)B且HB=BH,同时对于B的Sylow q-子群Q,只要满足(|H|,q)=1但(|H|,|Q^G|)≠1,便有HQ=QH,其中q是|B|的任一素因子.研究了τ-拟置换子群对有限群结构的影响.应用极小阶反例的方法得到了群G是p-超可解群的一个新的判定,又利用群G的F-剩余子群G^F的性质以及群G的准素数子群的τ-拟置换性得到了群G的半直积结构.A subgroup H of a finite group G is said to beτ-quasipermutable in G provided G has a subgroup B such that G=NG(H)B,HB=BHand H permutes with all Sylow q-subgroup Q of B such that(|H|,q)=1 but(|H|,|Q^G|)≠1,where q is any prime divisor of|B|In this paper,we investigate the influence ofτ-quasipermutable subgroups on the structure of finite groups We get a new criterion of p-supersolubility of G by using a way of the minimal counterexample.Moreover,the semidirect product structure of the group is obtained by using the properties of the F-residual subgroup of G and theτ-quasipermutability of some primary subgroups of G.

关 键 词：有限群 τ-拟置换子群 SYLOW子群 P-可解群 P-超可解群 HALL子群 

分 类 号：O152.1[理学—数学]