两类有限群的自同构群.pdf

文档介绍：Thesisforthe2009Master’SDegreeofShan)(iUniversityAutomorphismGroupsofTwoKindsfiniteGroupsNameSupervisorMajorFieldofResearchDepartmentResearchDurationQinGuoqiangAssociateProfessorHaoChengongFundamentalMathematicsGroupTheorySch001ofMathematicalSciencesSeptember,2006——June,2009June,2009万方数据目录引言??????????????????????????????1一预备知识???????????????????????????.4二主要结果及其证明???????????????????????..5三结论????????????????????????????.9四参考文献???????????????????????????lo五附录???????????????????????????..1l六致谢???????????????????????????..12万方数据Contents1Preliminaries????..?.??.????????????...cl2MainResultsand3Conclusion......................?......?.......?.....,................94References??????????????????????105Appendix??????????????????????.1l6Acknowledgments???????????????????.12万方数据中文摘要本文首先研究了齐次循环群的自同构群,,即可令G=G×?×G(r个),其中G=(9i)且o(9i)=n,i=1,2,?,,任取妒∈EndG,则可令9;『=9711?菸h,?,鳞=夕;”?鲧”,其中a巧∈Z。而1≤i,J≤/,口··?%、M(妒)=l???I∈M,(zn),\。一一n,,/则有环同构EndG笺M,(z。).,则AutG竺GL,(‰).,令n=p;1?p:。,其中诸鼽为两两不同的素数,则lAutG]=n:。兀;:,瞄“一巧勺1).其次,=NH可分解为正规子群Ⅳ与子群日的乘积时,我们定义了粘合自同构群Aut(G;N,H),相对自同构群Aut(G;NIH)以及相对外自同构群Out(G;N,H),得到下述结果:(0,仃)∈AutNXAutH为相容的自同构对,定义(ah)a=a。h9,Va∈N,h∈H,则OlEAutG,称为p和仃的粘合自同构,,如果Q∈AutG,则Q为粘合自同构当且仅当Q可正规化Ⅳ和日,即Ⅳ。=Ⅳ且H。==Ⅳ日,其中N司G,H≤G,则Aut(G;NIH)=Aut(G;N,H)=NH,其中NqG,H≤G,则Out(G;N\H、)竺Aut(G;N。H、)}NGLH丫。其中下::齐次循环群;自同构;自同态;粘合自同构群;相对自同构群万方数据ABSTRACTInthispaperwefirststudytheautomorphismgroupofahomocyclicgroup,=G×?×G(rtimes)beahomocyclicgroupwitha=(gi)ando(gi)=nfori=1,2,?,∈EndG,letg}=鲳11?9;h,?,够=g?”?簖”,wheren巧∈Zn,1≤i,J≤,=(兰:;;j三二)∈M,cz。,.ThenEndG竺M,(z。)(asaring)(zn)=pil?p;[=Ⅱ:1兀