【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.更多还原

【Abstract】 The definition of a type of the generalized Frattini subgroup aFrat(G) of infinite groups G is given,which is defined as the intersection of all abnormal maximal subgroups of G.The elementary properties of aFrat(G),especially the nilpotency of it,are studied.It turns out that in a finitely generated group G,the generalized Frattini subgroup aFrat(G) equals precisely to the subgroup generated by all abnormal eliminable subgroups of G and that in any group G,aFrat(G) coincides with the set of all abnormal nongenerators of G.It is also proved that local nilpotency,local solubility and local supersolubility are all aFrattini properties of the class of FC-groups,where the aFrattini property is a type of generalized Frattini property defined by aFrat(G).更多还原