关于有限群的Fsn-子群  

On F_(sn)-subgroups of finite groups

作　　者：马礼芳[1,2] 胡滨[1] 

机构地区：[1]江苏师范大学数学科学学院,江苏徐州221116 [2]除州广播电视大学铜山分校,江苏徐州221116

出　　处：《扬州大学学报（自然科学版）》2012年第3期1-4,共4页Journal of Yangzhou University：Natural Science Edition

基　　金：国家自然科学基金资助项目(11071229);江苏省高校自然科学基金资助项目(10KJD110004)

摘　　要：设F是一个群类.如果群G中存在一个正规子群T,使得HTG且(H∩T)HG/HG≤ZF∞(G/HG),则G的子群H称为G的Fsn-子群.利用Fsn-子群的概念得到Fsn-子群的性质以及可解群的一些新的判别准则,并对以前的结果进行推广.主要结论有:①设N是群G的非单位的正规子群,则N是可解群当且仅当G的每个不包含N的极大子群是G的Ssn-子群;②群G是可解群当且仅当G的每一个2-极大子群都是G的Ssn-子群;③设G是一个群,p是|G|的最小素因子,P是G的某个Sylowp-子群,则G是可解群当且仅当P的每个极大子群是G的Ssn-子群;④设G是一个群,p是|G|的最小素因子,P是G的某个Sylowp-子群.若G是A4-自由群且P的每个2-极大子群(如果存在)是G的Ssn-子群,则G是可解群.Let F be a class of groups. A subgroup H of a group G is said to be a Fsn-subgroup of G, if there exists a normal subgroup T of G such that HT is a subnormal subgroup of G and （H∩ T）HG/HG is contained in the F-hypercenter Z∞, （G/HG）. With the properties of Fsn-subgroups, some new criteria of solvable groups are given. Meanwhile the previous results are promoted. The main conclusions are as follows. ① Let N be a non-unit normal subgroup of G, then N is solvable group if and only if each maximal subgroup not containing N is an Fsn-subgroup of G. ② Group G is solvable if and only if every 2-maximal subgroup is an Fsn-subgroup of G. ③Let G be a group, p be the minimal prime factor of I GI and P be a Sylow p-group of G, then G is solvable if and only if every maximal subgroup of P is an Fsn-subgroup of G. ④ Let G be a group, p be the minimal prime factor of ︳G︳ and P be a Sylow p-subgroup of G. If G is an A4-free group and every 2-maximal subgroup of P （if existing） is an Fsn-subgroup of G, then G is solvable.

关 键 词：有限群 Fsn-子群 正规子群 可解群 

分 类 号：O152.1[理学—数学]