Posted on Categories:abstract algebra, 抽象代数, 数学代写

# 数学代写|抽象代数代写Abstract Algebra代考|MATH393 Definition 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|抽象代数代写Abstract Algebra代考|Definition and Examples

In this first case of study, a stream of ethanol is cooled, using water as cooling fluid. The process occurs in a counter-current heat exchanger, where ethanol enters the equipment at $60^{\circ} \mathrm{C}$, and it is desired to obtain an output temperature of $30^{\circ} \mathrm{C}$. The ethanol stream has a mass flow rate of $100 \mathrm{~kg} / \mathrm{h}$. The required flow rate of water $\left(w_{C}\right)$ is unknown, and it is directly related to the change of temperature of the water stream. Nevertheless, to obtain an initial design for the exchanger, a flow rate of $100 \mathrm{~kg} / \mathrm{h}$ is assumed for the cooling stream. A simplified representation of the process is shown in Figure 5.1.

In this chapter, we consider one of the most fundamental ideas of algebra-homomorphisms. The term homomorphism comes from the Greek words homo, “like,” and morphe, “form.” We will see that a homomorphism is a natural generalization of an isomorphism and that there is an intimate connection between factor groups of a group and homomorphisms of a group. The concept of group homomorphisms was introduced by Camille Jordan in 1870, in his influential book Traité des substitutions.
Definition Group Homomorphism
A homomorphism $\phi$ from a group $G$ to a group $G$ is a mapping from $G$ into $G$ that preserves the group operation; that is, $\phi(a b)=$ $\phi(a) \phi(b)$ for all $a, b$ in $G$.

Before giving examples and stating numerous properties of homomorphisms, it is convenient to introduce an important subgroup that is intimately related to the image of a homomorphism. (See property 4 of Theorem 10.1.)

## 数学代写|抽象代数代写Abstract Algebra代考|Properties of Homomorphisms

Let $\phi$ be a homomorphism from a group $G$ to a group $G$ and let $g$ be an element of $G$. Then

$\phi$ carries the identity of $G$ to the identity of $G$.

$\left.\phi\left(g^{n}\right)=\phi(g)\right)^{n}$ for all $\boldsymbol{n}$ in $Z$.

If $|g|$ is finite, then $|\phi(g)|$ divides $|g|$ and if $|G|$ is finite then $|\phi(g)|$ divides $|g|$ and $|\phi(G)|$.

Ker $\phi$ is a subgroup of $G$.

$\phi(a)=\phi$ (b) if and only if aKer $\phi=b \operatorname{Ker} \phi$.

If $\phi(g)=g^{\prime}$, then $\phi^{-1}\left(g^{\prime}\right)=\left{x \in G \mid \phi(x)=g^{\prime}\right}=g \operatorname{Ker} \phi$.
PROOF The proofs of properties 1 and 2 are identical to the proofs of properties 1 and 2 of isomorphisms in Theorem 6.2. To prove property 3 , notice that properties 1 and 2 together with $g^{n}=e$ imply that $e=\phi(e)=\phi\left(g^{n}\right)=(\phi(g))^{n}$. So, by Corollary 2 to Theorem 4.1, we have $|\phi(g)|$ divides $n$. That $|\phi(g)|$ divides $|\phi(G)|$ when $G \mid$ is finite follows from Lagrange’s theorem.
By property 1 we know that $\operatorname{Ker} \phi$ is not empty. So, to prove property 4 , we assume that $a, b \in \operatorname{Ker} \phi$ and show that $a b^{-1} \in \operatorname{Ker} \phi$. Since $\phi(a)=e$ and $\phi(b)=e$, we have
$\phi\left(a b^{-1}\right)=\phi(a) \phi\left(b^{-1}\right)=\phi(a)(\phi(b))^{-1}=e e^{-1}=e$. So,
$a b^{-1} \in \operatorname{Ker} \phi$.

## 数学代写|抽象代数代写Abstract Algebra代考|Definition and Examples

$\mathrm{A}$ 同态 $\phi$ 从一组 $G$ 对一组 $G$ 是来自的映射 $G$ 进入 $G$ 保留组操作; 那隄, $\phi(a b)=\phi(a) \phi(b)$ 对所有人 $a, b$ 在 $G$.

## 数学代写|抽象代数代写Abstract Algebra代考|Properties of Homomorphisms

$\phi$ 菷有身份G以身份 $G .$
$\left.\phi\left(g^{n}\right)=\phi(g)\right)^{n}$ 对所有人n在 $Z$.

$\phi(a)=\phi(\mathrm{b})$ 当且仅当 $\mathrm{aKer} \phi=b \operatorname{Ker} \phi .$

$\phi\left(a b^{-1}\right)=\phi(a) \phi\left(b^{-1}\right)=\phi(a)(\phi(b))^{-1}=e e^{-1}=e$. 所以，
$a b^{-1} \in \operatorname{Ker} \phi$.

## 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。