Posted on Categories:Topology, 拓扑学, 数学代写

# 数学代写|拓扑学代写TOPOLOGY代考|MATH611 Direct Products of Groups

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|拓扑学代写TOPOLOGY代考|Direct Products of Groups

If the theme of the previous section was finding smaller groups within bigger groups, the theme of this section is constructing bigger groups from smaller ones. There are several such constructions in group theory, and we will present only one of thesewhich is the most straightforward and ubiquitous of such constructions.

Let $G_1$ and $G_2$ be two groups. We will show that the direct product of $G_1$ and $G_2$, namely the set of pairs of elements, one from $G_1$ and one from $G_2$, defined by
$$G_1 \times G_2=\left{\left(g_1, g_2\right): g_1 \in G_1 \text { and } g_2 \in G_2\right},$$
can be made into a group. To do this, we first must define a binary operation on $G_1 \times G_2$. The obvious choice is

$$\left(g_1, g_2\right) \cdot\left(h_1, h_2\right)=\left(g_1 h_1, g_2 h_2\right)$$
for any $\left(g_1, g_2\right)$ and $\left(h_1, h_2\right)$ in $G_1 \times G_2$. Next, we must define an identity element, and the obvious choice is
$$e=\left(e_1, e_2\right),$$
where $e_1$ is the identity in $G_1$ and $e_2$ is the identity in $G_2$. Finally, we must define inverses, and the obvious choice is
$$\left(g_1, g_2\right)^{-1}=\left(g_1^{-1}, g_2^{-1}\right)$$
for every $\left(g_1, g_2\right) \in G_1 \times G_2$.

## 数学代写|拓扑学代写TOPOLOGY代考|Homomorphisms

A general principle in mathematics is that once you have defined an interesting structure, you should also study the maps that preserve that structure. Thus when you are studying topology, you should study continuous functions and especially homeomorphisms. We now consider the types of maps between groups that preserve the basic structure. These are known as homomorphisms; the homomorphisms that are bijective are called isomorphisms.

Remark 6.6 Do not get homomorphisms confused with homeomorphisms. Despite the similarities in the words, they are very different notions. Homomorphisms are for groups, or more generally for algebraic structures, whereas homeomorphisms are for topological spaces. In fact, homeomorphisms of topological spaces more closely resemble isomorphisms of groups. We will see that there are relationships of this type once we have studied our group-theoretic invariants of topological spaces.
Definition 6.7 Let $(G, \cdot)$ and $\left(G^{\prime}, \otimes\right)$ be two groups. Then a function $f: G \rightarrow G^{\prime}$ is said to be a homomorphism if for any $g_1, g_2 \in G$ we have
$$f\left(g_1 \cdot g_2\right)=f\left(g_1\right) \otimes f\left(g_2\right) .$$

## 数学代写|拓扑学代写TOPOLOGY代考|Direct Products of Groups

《left 缺少或无法识别的分隔符

$$\left(g_1, g_2\right) \cdot\left(h_1, h_2\right)=\left(g_1 h_1, g_2 h_2\right)$$

$$e=\left(e_1, e_2\right)$$

$$\left(g_1, g_2\right)^{-1}=\left(g_1^{-1}, g_2^{-1}\right)$$

## 数学代写|拓扑学代写TOPOLOGY代考|Homomorphisms

$$f\left(g_1 \cdot g_2\right)=f\left(g_1\right) \otimes f\left(g_2\right) .$$

## 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。