群的融合自由积的两种广义Frattini子群

摘要

M.K.Azarian generalized a theorem on Frattini subgroups of amalgamated free products of two subgroups proposed by C.Y.Tang to the case of the free product of any set of subgroups with amalgamated subgroup.Now as a first step two types of generalized Frattini subgroups,psn Frattini subgroups and psc Frattini subgroups are introduced.They are defined as the intersection of all maximal normal subgroups and the intersection of all maximal characteristic subgroups respectively.Then psn Frattini subgroups and psc Frattini subgroups of the free product of any set of subgroups with amalgamated subgroup being cyclic are considered from two different points of view and similar results are obtained,which are a generalization of the results of M.K.Azarian and Q.Guo.%Azarian将Tang得到的关于两个群的带循环融合自由积的Frattini子群的一个定理推广到任意多个子群的带循环融合自由积的情形.文章首先引入任意群G的两种广义Frattini子群,psn Frattini子群和psc Frattini子群,它们分别定义为指数为素数方幂ps的极大正规子群的交和指数为素数方幂ps的极大特征子群的交,其中p为任意素数.然后分别从两个不同角度出发,考虑任意多个子群的带循环融合自由积的psn Frattini子群和psc Frattini子群,得到了类似的结果,从而推广了Azarian和郭钦等人的结果.