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中文摘要：
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.
英文摘要：
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.
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The structure of finite non-nilpotent groups in which every 2-maximal subgroup permutes with all 3-m
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