非正规循环子群的正规化子皆极大的两类有限可解群

Two Finite Solvable Groups in Which the Normalizer of Some Non-Normal Subgroups is Maximal

子群的正规性和有限群的结构有密切的关系,而正规化子作为子群正规性的一种度量对有限群结构的影响自然也很大.极大子群是有限群的一类重要子群.利用某些子群的正规化子的极大性研究有限群的结构.具体研究了群G的阶被p整除的非正规循环子群的正规化子皆极大的有限可解群,以及非正规p-子群和{p,q}-子群的正规化子均极大的有限可解群.得到这两类群的一些性质,并对这两类群的结构给出了刻画.

The normality of subgroups is closely related to the structure of finite groups,and the normalizer of subgroups,which is a measure of the normality of subgroups,has a significant influence on their structure.On the other hand,the maximal subgroup is an important kind of subgroup of finite groups.So it is reasonable to investigate the structure of a group by using normalizers of some kind of subgroups.In this paper,we study the solvable groups in which the normalizer of cyclic subgroups whose order is divided by pis maximal in G.We also study the solvable groups in which every non-normal p-subgroup and{p,q}-subgroup have a maximal normalizer in G.Some good properties are given for the above two types of group,and we also describe the structure of the two types of group.