cc子群，极小子群的c可补性对有限群结构的影响

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1、CC．子群等对有限群结构的影响中文摘要cc一子群，极小子群的C一可补性对有限群结构的影响中文摘要设G是有限群，H≤G，称日为G的一个CC-子群，如果对任意的1≠z∈H，都有Cc(x)≤H，记为H≤G．若H≠G，则记为日

2、规，则(1)H幂零且在G中有补也为CC-子群．(2)G为Frobenius．群．其次，本文利用极小子群的c一可补性来研究有限群的可解性与p一幂零性．郭秀云和岑嘉评在文[15】中证明了：设P是群G阶的最小素因子，尸是G的Sylowp一子群，如果PnGw的每一极小子群在G中c一可朴，且当P=2时或者PnGⅣ的每一4阶循环子群在G中c一可补或者P与四元数群无关，则G是p一幂零的，这里G∥是G的幂零剩余．本文通过“假设子群在G的子群G’中c一可补”，证明了G是可解的；“假设子群在Op(G)中c．可补”，证明了G是p一幂零的．

3、关键词；CC一子群，Frobenius．群，c一可补子群，p一幂零群，可解群．作者：高培培指导老师：施武杰(教授)CC-子群等对有限群结构的影响AbstractTheinfluenceofee—subgroups，c-supplementarityotminimal一．一，．‘’subgroupsonthestructureoflinitegroups^⋯AbstractLetHbeasubgroupofafinitegroupG．HiscalledaCC—subgroupofG，ifforany1≠z∈H，C『G(

4、z)≤H．LetGbeafinitegroup．AsubgroupHofGissaidtobec—supplementedinGiftheree)(istsasubgroupKofGsuchthatG=HK，HnK≤Ha=Corer(H)．Firstly,thispaperintroducessomepropertiesonoc—subgroupsandanddiscussesthefinitegroupswhichcontainpropercc-subgroup．UnderthefamoustheoremofSch

5、ur．Zassenhaus，weprovethefollowingtheorem：LetHbeasolvableandnormalCC—subgroupofG．Then(1)HisnilpotentandhasacomplementinGandallsuchcomplementsaxeCC-subgroupsofG．(2)GisaFrobeniusgroup．Secondly,inthispaper，westudythesolvabilityandP—nilpotencyoffinitegroupsunderthea

6、ssumptionthatsomeminimalsubgroupsarec-supplemented．In[15]，GuoandShumprovedthefollowingtheorem：LetGbeafinitegroupandLetPbeaSylowp-subgroupofGwherePisaminimalprimedisvisorofIa1．SupposethateveryminimalsubgroupofPnGNisc—supplementedinG，andwhenP=2，eithereverycyclics

7、ubgroupoforder4ofPnGNisc-supplementedinGorPisquaternion·free，whereGⅣistheⅣ．residueofG．InthispaperweprovethatGissolvableundertheassumptionthatthesubgroupisc-supplementedinG’．WealsoproveGisP—nilpotentundertheassumptionthatthesubgroupisc-supplementedinOp(G)．Keywor

8、ds：CC—subgroups：Frobenius—group；c—supplementedgroup；solv-ablegroup；p-nilpotent．WrittenbyGaoPeipeiSupervisedbyProf．ShiWujieII苏州大学学位论文独创性声明及使用授权声明学位论文独创性声明本人郑重声明：所提交的学位论文是本人在导