无限群的伪正规极大子群的交  

The Intersection of Abnormal Maximal Subgroups of Infinite Groups

作　　者：张志让[1] 杨四强[1] 李雪梅[1] 

机构地区：[1]成都信息工程学院数学学院,四川成都610225

出　　处：《四川师范大学学报（自然科学版）》2013年第3期350-355,共6页Journal of Sichuan Normal University（Natural Science）

基　　金：国家自然科学基金(11071229)资助项目

摘　　要：在无限群中定义了另一种广义Frattini子群aFrat(G),它等于群G的所有伪正规极大子群的交.研究了aFrat(G)的基本性质,讨论了aFrat(G)的幂零性,指出在有限生成群G中,aFrat(G)恰等于G的所有伪正规多余子群生成的子群,证明了群G中,aFrat(G)恰由G的某种非生成元构成,并且在FC-群中证明了局部幂零性、局部可解性和局部超可解性都是aFrattini性质,其中,aFrattini性质是由aFrat(G)定义的广义Frattini性质.The definition of a type of the generalized Frattini subgroup ,,Frat（G） of infinite groups G is given, which is defined as the intersection of all abnormal maximal subgroups of G. The elementary properties of ,,Frat （G） , especially the nilpotency of it, are studied. It turns out that in a finitely generated group G, the generalized Frattini subgroup ,,Frat（G） equals precisely to the subgroup generated by all abnormal eliminable subgroups of G and that in any group G, αFrat（G） coincides with the set of all abnormal nonge- nerators of G. It is also proved that local nilpoteney, local solubility and local supersolubility are all ,,Frattini properties of the class of FC-groups, where the αFrattini property is a type of generalized Frattini property defined by αFrat（G）.

关 键 词：伪正规极大子群 伪正规非生成元 aFrattini子群 aFrattini性质 FC-群 

分 类 号：O152.1[理学—数学]