抽象代数学习心得

TheLearningExperienceOfAbstractAlgebra抽象代数学习心得WhenIcontactedwithabstractalgebrafirstly,Ifeltlikesuchacoursewasverydifficultforme,becausethematerialiswritteninEnglish,eachonestrangeEnglishwordbroughtmealotofpressure.Especiallyintheclass,IfeelthatIcan'tkeepupwiththeteacher.Becausebeforeunstandingthedefinitionduringmystudy,IhavetotranslatetheEnglishwordsbacktotheChineseinmymind,soitgreatlyreducedtheefficiencyofmystudyandithasbecomeoneofthebiggestdifficultiesinmylearningabstractalgebra.当我刚开始接触抽象代数这么课程时，我感觉这么课程对我来说是很困难的，因为教材是全英文撰写的，一个个陌生的英语词汇给我带来了很大的压力。尤其在课堂上，我感觉我完全不能跟上老师思路。因为我在学习过程中在理解和思考定义之前，我必须将英文词汇的意思在脑海中翻译回中文，这样大大地降低了我学习的效率，因此成了我学习抽象代数中的最大困难之一。WhenIwasthinkingabouthowtosolvethedifficulties,Ithinkbacktothereferencebookswhichtheteacherhadrecommendedtous,soIfoundsomereferencebooksaboutabstractalgebraintheschoollibrary.Afterreadingthesebooks,theymakemefeelrelaxedstudyingofabstractalgebra.BecausethesereferencebooksareinChineseandtheyeliminatedtheambiguityofunderstandingthedefinitionortheoremwhichcausedbyIwasnotfamiliarwiththeEnglish.Beforeclass,IwillseeaChinesereferencebookfirst,andthenlookingattheteachingmaterialwhichwritteninEnglish,itwillmakemefeelmucheasiertounderstandtheteachingmaterialcontent.在我思考怎样解决这个困难的时候，我回想到老师向我们推荐的参考书，于是我在学校图书馆找到了一些关于抽象代数的参考书。阅读这些参考书之后，使我感觉抽象代数的学习变得轻松了些，因为这些参考书是中文的，消除了因对英文的不熟悉而引起对定义或定理理解的歧义。在上课前的预习，我都会先看一次中文的参考书，再看全英的教材，使我感觉对教材上的内容的理解也变得轻松了些。Aftertwomonthsoflearning,Ihavelearntthatabstractalgebraismainlydoingresearchesonalgebraicstructureonthebasicofthesetandmapping.Inthefirstchapter,wemainlystudythedefinitionandrepresentationofsets,therelationshipbetweenthesets,theoperationofsetandmappingandsoon.Thisissimilarwiththatcontentoftheadvancedalgebra.Inthefunctionstudy,weneedtodistinguishtheinjective,bijectiveandsurjectiveclearly.Andwhenthefunctionfisbothinjectiveandsurjective,whichiselementsofasettoelementsofanothersetisone-to-one,sowecansaidthatthefunctionfisbijective,anditistheidenticaltransformationofadvancedalgebra.Wenotonlystudytherelationshipbetweenthesets,butalsostudytherelationshipbetweenelementsofaset,includingtheidentityrelationshipsandpartitionofaset.经过两个月的学习后，我了解到抽象代数主要在集合和映射的基础上研究各种代数结构。在第一章里，我们主要学习集合的定义、表示方法、集合之间的关系、集合的运算法则和映射、这些与高等代数的内容很相似。其中在函数的学习中，需要把单射，双射和满射区分清楚。而且当函数f满足单射和双射时，一个集合的元素到另一个集的元素合是一一对应的，这个函数是满射的，并且就是高等代数中的恒等变换。我们不但要研究集合之间的关系，而且还研究了集合的元素之间的关系，包括集合中的恒等关系和划分。Inthesecondchapter,wemainlystudiedbinaryoperation,group,subgroup,commutativegroupandsoon.Thebinaryoperationisthemostcommonoperations,suchasvariousoperationsofaddition,subtraction,multiplication,anddivisionbetweenobjects.What’smore,thebinaryoperationisoneofthebasicelementsofagroupwhichismadeupofasetandabinaryoperation.Toqualifyasanabeliangroup,thesetandoperationmustbesatisfiedfiverequirementsknownastheabeliangroupaxioms:closure,associativity,identityelement,inverseelementandcommutativity.AgroupGhasexactlyoneidentityelemente,andanelementxbelongtoGmustexistaninverseelementtomakethatxx’equalstoe.Therefore,agroupalsosatisfiesthecancellationlawwhichisoneoftheelementarypropertiesofgroups.Learningandunderstandingthedefinitionandpropertionofgroupisthefoundationoflearningtheknowledgeofsecondchapter.Duringstudyinggrouptheorywhichhascertainabstractness,wecanlearncombiningwithexamplesinordertolearntheknowledgewell.Becausethattheorycombinedwiththepracticalproblems,theycanmaketheabstractcontentintoconcreteimage.在第二章中，主要学习了二元运算、群、子群、交换群等。其中二元运算是最常见的运算，比如各种对象之间的加减乘除运算。更是构成群的基本元素之一，群是由一个集合和一个二元运算构成，群的集合元素运算必须满足封闭性和结合律。群具有唯一的单位元，而且一个元素X必须存在一个逆元，使得X*X=e，以及消去律，这是群的基本性质。学习群和理解群的定义与性质，是学习第二章内容的基础，在学习群的理论中，群的理论具有一定的抽象性，所以为了更好地理解可以结合例题学习。SuchasletHandKbesubgroupofagroupG.ThenHUKisalsoasubgroupofG?Ibelievethatmanypeoplewouldsayyes,butweuseaexample,like:G=（Z，+）isagroupandnisanyinteger.ThenthesetnZ=nofmultiplesofnformsasubgroupofG.2ZU6ZisasubgroupofG,since2ZU6Z=2Z.anohter2ZU3Zisnotasubgroupof(Z,+).forexample,3+2=5notbelongto2ZU3Zsoisnotclosedunderaddition.Whenwecantryingafewmoreexamples,wecanfoundnZUn’ZissubgroupofnZorn,Z,thennZUn’ZissubgroupofG.Sowecan'ttreatabstractalgebraproblemwithhabitsofthinking,Therefore,wemusttothinkseriouslyaboutthetitle.Theoneforthecaselawswhichusedinthejudgementofabstractalgebrapropositionsarefrequently.Intheprocessofbuildingacounterexamplecanexerciseourabilitytoimagineverywell.例如:LetHandKbesubgroupofagroupG.ThenHUKisalsoasubgroupofG?我相信很多人会说是，但是我们举个例子，例如：G=（Z，+）isagroupandnisanyinteger.ThenthesetnZ=nofmultiplesofnformsasubgroupofG.2ZU6ZisasubgroupofG,since2ZU6Z=2Z.anohter2ZU3Zisnotasubgroupof(Z,+).forexample,3+2=5notbelongto2ZU3Zsoisnotclosedunderaddition.当我们举更多的例子时，我们可以发现当nZUn’Z是nZ或n’Z的其中一个时，nZUn’Z才是G的子群。所以我们不能用以前的思维习惯去对待抽象代数的问题，因此我们必须对题目进行思考。其中举反例法在判断抽象代数的命题中比较常用。在构建反例的过程中能很好地锻炼我们的想象能力。Inlearningcyclicgroup,symmetrygroup,cosetandnormalsubgroup,Ithinkthatweneedtopayattentiontothese.Cyclicgrouphaveoneelementa,makesanyoneelementofGcanbecomposedofthepowerelementa,thebinaryoperationofcyclicgroupistosatisfythecommutativelaw,soanyacyclicgroupmustbeAbeliangroup.ButitisnotnecessarilythattheAbeliangroupiscycli