【摘要】 设G是一个有限群,F是一个群类.如果存在G的一个正规子群T使得HT是G的正规子群,并且(H∩T)HG/HG包含在G/HG的F-超中心ZF∞(G/HG)中,则称G的子群H在G中Fn-正规.利用Fn-正规子群的性质给出超可解群和可解群的一些新的判别准则,并对以前的结果进行推广.主要定理有:①设G是一个可解群,G超可解当且仅当G的每个次正规子群在G中Un-正规.②设G是一个有限群,N是G的一个非平凡正规子群,则N可解当且仅当G的每个不包含N的极大子群在G中Sn-正规.③群G是可解的当且仅当下列两个条件之一满足:(a)存在G的Sylow 2-子群P使得P的每个极大子群在G中Sn-正规;(b)对G的某个Sylow 2-子群,P在G中Sn-正规.更多还原

【Abstract】 Let G be a finite group and F a class of groups.A subgroup H of G is Fn-normalin G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and(H∩T)HG/HG is contained in the F-hypercenter ZF∞(G/HG) of G/HG.Using Fn-normal subgroup some new characterizations of some classes of groups are given and a series of known results are generalized.① Let G be a soluble group.Then G is supersoluble if and only if every subnormal subgroup of G is Un-normal in G.② Let G be a group and N a non-identity normal subgroup of G.Then N is soluble if and only if every maximal subgroup of G not containing N is Sn-normal in G.③ A group G is soluble if and only if one of the following conditions holds:(a) There exists a Sylow 2-subgroup P of G such that every maximal subgroup of P is Sn-normal in G.(b) There exists a Sylow 2-subgroup P of G such that every maximal subgroup of P is Sn-normal in G.更多还原