Posted on Categories:Number Theory, 数学代写, 数论

# 数学代写|数论代写Number Theory代考|MATH3320 Definitions, basic properties, and examples

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|数论代写Number Theory代考|Definitions, basic properties, and examples

Definition 8.1. An abelian group is a set $G$ together with a binary operation $\star$ on $G$ such that
(i) for all $a, b, c \in G, a \star(b \star c)=(a \star b) \star c$ (i.e., $\star$ is associative),
(ii) there exists $e \in G$ (called the identity element) such that for all $a \in G, a \star e=a=e \star a$,
(iii) for all $a \in G$ there exists $a^{\prime} \in G$ (called the inverse of a) such that $a \star a^{\prime}=e=a^{\prime} \star a$,
(iv) for all $a, b \in G, a \star b=b \star a$ (i.e., $\star$ is commutative).
While there is a more general notion of a group, which may be defined simply by dropping property (iv) in Definition $8.1$, we shall not need this notion in this text. The restriction to abelian groups helps to simplify the discussion significantly. Because we will only be dealing with abelian groups, we may occasionally simply say “group” instead of “abelian group.”

## 数学代写|数论代写Number Theory代考|Subgroups

We next introduce the notion of a subgroup.
Definition 8.4. Let $G$ be an abelian group, and let $H$ be a non-empty subset of $G$ such that
(i) $a+b \in H$ for all $a, b \in H$, and
(ii) $-a \in H$ for all $a \in H$.
Then $H$ is called a subgroup of $G$.
In words: $H$ is a subgroup of $G$ if it is closed under the group operation and taking inverses.

Multiplicative notation: if the abelian group $G$ in the above definition is written using multiplicative notation, then $H$ is a subgroup if $a b \in H$ and $a^{-1} \in H$ for all $a, b \in H$.

Theorem 8.5. If $G$ is an abelian group, and $H$ is a subgroup of $G$, then $H$ contains $0_G$; moreover, the binary operation of $G$, when restricted to $H$, yields a binary operation that makes $H$ into an abelian group whose identity is $0_G$.

Proof. First, to see that $0_G \in H$, just pick any $a \in H$, and using both properties of the definition of a subgroup, we see that $0_G=a+(-a) \in H$. Next, note that by property (i) of Definition $8.4, H$ is closed under addition, which means that the restriction of the binary operation “+” on $G$ to $H$ induces a well defined binary operation on $H$. So now it suffices to show that $H$, together with this operation, satisfy the defining properties of an abelian group. Associativity and commutativity follow directly from the corresponding properties for $G$. Since $0_G$ acts as the identity on $G$, it does so on $H$ as well. Finally, property (ii) of Definition $8.4$ guarantees that every element $a \in H$ has an inverse in $H$, namely, $-a$.

## 数学代写|数论代写数论代考|定义、基本属性和实例

(ii) 在G$中存在$e（称为身份元素），这样对于G中的所有$a，a\star e=a=e\star a$。
(iii) 对于所有$a\in G$，存在$a^{\prime}\in G$（称为a的逆），使得$a\star a^{\prime}=e=a^{prime}\star a$。
(iv) 对于G中的所有$a, b\，a\star b=b\star a$（即$star$是换元的）。

## 数学代写|数论代写数字理论代考|子群

(i) $a+b\in H$为所有$a, b\in H$，并且
(ii) $-a\in H$为所有$a\in H$。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。