抽象代数学习心得

抽象代数学习心得 -CAL-FENGHAI-(2020YEAR-YICAI)_JINGBIAN
The Learning Experience Of Abstract Algebra
抽象代数学习心得
When I contacted with abstract algebra firstly,I felt like such a course was very difficult for me, because the material is written in English, each one strange English word brought me a lot of pressure. Especially in the class, I feel that I can't keep up with the teacher. Because before unstanding the definition during my study, I have to translate the English words back to the Chinese in my mind, so it greatly reduced the efficiency of my study and it has become one of the biggest difficulties in my learning abstract algebra.
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。

When I was thinking about how to solve the difficulties, I think back to the reference books which the teacher had recommended to us, so I found some reference books about abstract algebra in the school library. After reading these books, they make me feel relaxed studying of abstract algebra. Because these reference books are in Chinese and they eliminated the ambiguity of understanding the definition or theorem which caused by I was not familiar with the English. Before class, I will see a Chinese reference book first, and then looking at the teaching material which written in English, it will make me feel much easier to understand the teaching material content.
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。

After two months of learning, I have learnt that abstract algebra is mainly doing researches on algebraic structure on the basic of the set and mapping. In the first chapter, we mainly study the definition and representation of sets, the relationship between the sets, the operation of set and mapping and so on. This is similar with that conten t of the advanced algebra. In the function study, we need to distinguish the injective, bijective and surjective clearly. And when the function f is both injective and surjective, which is elements of a set to elements of another set is one-to-one, so we can said that the function f is bijective, and it is the identical transformation of advanced algebra. We not only study the relationship between the sets, but also study the relationship between elements of a set, including the identity relationships and partition of a set.
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。

In the second chapter, we mainly studied binary operation, group, subgroup, commutative group and so on . The binary operation is the most common operations, such as various operations of addition, subtraction, multiplication, and division between objects. What’s more, the binary operation is one of the basic elements of a group which is made up of a set and a binary operation. To qualify as an abelian group , the set and operation must be satisfied five requirements known as the abelian group axioms: closure, associativity, identity element, inverse element and commutativity.
A group G has exactly one identity element e, and an element x belong to G must exist an inverse element to make that xx’ equals to e. Therefore, a group also satisfies the cancellation law which is one of the elementary properties of groups. Learning and understanding the definition and propertion of group is the foundation of learning the knowledge of second chapter. During studying group theory which has certain abstractness, we can learn combining with examples in order to learn the knowledge well. Because that theory combined with the practical problems, they can make the abstract content into concrete image.
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。

Such as let H and K be subgroup of a group G. Then HUK is also a subgroup of G I believe that many people would say yes, but we use a example, like:
G=（Z，+）is a group and n is any integer. Then the set nZ=<n> of multiples of n forms a subgroup of G.2Z U6Z is a subgroup of G, since 2Z U6Z=2Z.anohter 2ZU3Z is not a subgroup of (Z,+).for example, 3+2=5 not belong to 2Z U3Z so is not closed under addition. When we can trying a few more examples, we can found nZ U n’Z is subgroup of nZ or n,Z, then nZ U n’Z is subgroup of G.
So we can't treat abstract algebra problem with habits of thinking, Therefore, we must to think seriously about the title.The one for the case laws which used in the judgement of abstract algebra propositions are frequently. In the process of building a counterexample can exercise our ability to imagine very well.
例如:Let H and K be subgroup of a group G. Then HUK is also a subgroup of G 我相信很多人会说是，但是我们举个例子，例如：
G=（Z，+）is a group and n is any integer. Then the set nZ=<n> of multiples of n forms a subgroup of G.2Z U6Z is a subgroup of G, since 2Z U6Z=2Z.anohter 2ZU3Z is not a subgroup of (Z,+).for example, 3+2=5 not belong to 2Z U3Z so is not closed under addition. 当我们举更多的例子时，我们可以发现当nZ U n’Z是nZ或n’Z的其中一个时，nZ U n’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。

In learning cyclic group, symmetry group, coset and normal subgroup, I think that we need to pay attention to these. Cyclic group have one element a, makes any one element of G can be composed of the power element a , the binary operation of cyclic group is to satisfy the commutative law, so any a cyclic group must be Abelian group. But it is not necessarily that the Abelian group is cyclic group, such as G = (Z, +). In addition, they must learn to find generating element when we know the binary operation of the cyclic group. In the symmetry group, we not only know the operation which is about two families of group left group start from right to, but also understand the mean that symmetry group is how to show in the family. Beside, we should know the difference between isomorphism and homomorphism. Isomorphism is that the mapping of a set to another set is a bijection, and homomorphic is that mapping f is injective or bijection, if a homomorphic mapping f satisfies set V is mapped to the same set V, at this time it also is homogeneous, and f is the automorphism or endomorphism on set V.
在学习循环群、对称群、陪集和正规子群时，我认为我们需要注意这些循环群存在一个元a，使得G中的任何一个元素都能由a的幂组成，则循环群的二元运算是满足交换律的，所以任何一个循环群必定是阿贝尔群。

但是阿贝尔群却不一定是循环群，如 G=（Z，+）。

除此之外，还必须学会根据循环群的二元运算找出生成元。

而在对称群中，群的两个族的运算是从右到左的，还要了解对称群中的族是如何表示的。

还要区别同构与同态，同构是一个集合到另一个集合的映射是双射，而同态映射f是单射或双射的，若同态映射f满足一个集合V映射到相同的集合V中时，此时亦是同构的，并称f是V的自同构或自同态。

So learning abstract algebra needs to understand each definition of abstract algebra, and it is combined with the actual examples, to learn. in this way we can feel the knowledge that the abstract become concrete, in addition to thinking, we also have to review and finish the homework in time, because of abstract algebra this course is based on the English course, and the content is more. If we not timely review will be easy to forget . Learning to analyze differences between each definition,it can deepen our understanding. That is some of the tips that I learn abstract algebra.
所以学习抽象代数需要理解每个定义，并且是结合实际的例题去学习，这样才能把抽象化成具体的知识去感受，除了思考以外，我们还必须要及时复习和完成课后作业，因为抽象代数这么课程是英语为基础的课程，而且内容较多。

如不及时复习会容易遗忘。

还要学会分析各个定义之间的不同点，这样才能加深理解。

以上就是我学习抽象代数的一些心得体会。