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# 数学代写|代数拓扑代考Algebraic Topology代考|Exact Sequence of Groups

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## 数学代写|代数拓扑代考Algebraic Topology代考|Exact Sequence of Groups

This section conveys some results of exact sequences of groups and their homomorphisms which are frequently applied in algebraic topology. For this section the book Adhikari and Adhikari (2014) is referred.
Definition 1.4.1 A sequence of groups and their homomorphisms
$$\cdots \rightarrow G_{n+1} \stackrel{f_{n+1}}{\longrightarrow} G_n \stackrel{f_n}{\longrightarrow} G_{n-1} \rightarrow \cdots$$
is said to be exact if ker $f_n=\operatorname{Im} f_{n+1}$ for all $n$. Clearly, $f_n \circ f_{n+1}=0$ for an exact sequence.

Proposition 1.4.2 (i) In the short exact sequence $0 \rightarrow G \stackrel{f}{\longrightarrow} K$, $f$ is $a$ monomorphism.
(ii) In the short exact sequence $G \stackrel{f}{\longrightarrow} K \rightarrow 0, f$ is an epimorphism.
(iii) The sequence $0 \rightarrow G \stackrel{f}{\longrightarrow} K \rightarrow 0$ is exact if and only if $f$ is an isomorphism;
(iv) If $G$ is a normal subgroup of $K$ and i : $\hookrightarrow K$ is the inclusion map (i.e., $i(x)=x$ for all $x \in G)$, then the sequence
$$0 \rightarrow G \stackrel{i}{\longrightarrow} K \stackrel{p}{\longrightarrow} K / G \rightarrow 0$$
is an exact sequence, where 0 denotes the trivial group and $p$ is the natural homomorphism defined by $p(x)=x+G$ for all $x \in K$.

Proposition 1.4.3 Given exact sequences of groups and homomorphisms
$$0 \rightarrow G_i \stackrel{f_i}{\longrightarrow} K_i \stackrel{g_i}{\longrightarrow} H_i \rightarrow 0$$
for each element $i \in I$, the sequence
$$0 \rightarrow \bigoplus_{i \in I} G_i \stackrel{\oplus f_i}{\longrightarrow} \bigoplus_{i \in I} K_i \stackrel{\oplus g_i}{\longrightarrow} \bigoplus_{i \in I} H_i \rightarrow 0$$
is also exact

## 数学代写|代数拓扑代考Algebraic Topology代考|Free Product and Tensor Product of Groups

This subsection conveys the concept of free product of groups.
Definition 1.5.1 Let $G$ and $H$ be groups (not necessarily abelian). Their free product denoted by $G * H$ is a group satisfying the following condition: if there are homomorphisms $i$ and $j$ such that given a pair of homomorphisms $f: G \rightarrow K$ and $g: H \rightarrow K$ for any group $K$, there exists a unique homomorphism $h: G * H \rightarrow K$ making the diagram in Fig. 1.2 commutative.
For example, $\mathbf{Z} * \mathbf{Z}$ is a free group (of rank 2 ).

Remark 1.5.2 An alternative description of free product $G * H$ may be given with the help of presentations of groups $G$ and $H$.

Definition 1.5.3 Let $G=\langle X: R\rangle$ and $H=\langle Y: S\rangle$ be presentations of the groups $G$ and $H$ in which the sets $X$ and $Y$ are generators (and thus the relations $R$ and $S$ ) are disjoint. Then a presentation of $G * H$ is given by
$$G * H=\langle X \cup Y: R \cup S\rangle$$

This subsection conveys the concept of tensor products of groups.
Definition 1.5.4 Let $G$ and $H$ be two abelian groups. Their tensor product denoted by $G \otimes H$ is the group defined as the abelian group generated by all pairs of the form $(g, h)$ with $g \in G, h \in H$ satisfying the bilinearity relations $\left(g+g^{\prime}, h\right)=(g, h)+$ $\left(g^{\prime}, h\right)$ and $\left(g, h+h^{\prime}\right)=(g, h)+\left(g, h^{\prime}\right)$.

For example, $\mathbf{Z}m \otimes \mathbf{Z}_n=\mathbf{Z}{(m, n)}$, where $(m, n)$ is the gcd of $m$ and $n$ : on the other hand, this tensor product is 0 , if $m$ and $n$ are relatively prime.

## 数学代写|代数拓扑代考Algebraic Topology代考|Exact Sequence of Groups

1.4.1群及其同态的序列
$$\cdots \rightarrow G_{n+1} \stackrel{f_{n+1}}{\longrightarrow} G_n \stackrel{f_n}{\longrightarrow} G_{n-1} \rightarrow \cdots$$

(二)在短的确切序列$G \stackrel{f}{\longrightarrow} K \rightarrow 0, f$是一个外胚。
(iii)序列$0 \rightarrow G \stackrel{f}{\longrightarrow} K \rightarrow 0$是精确的当且仅当$f$是同构的;
(iv)如果$G$是$K$的正子群，并且i: $\hookrightarrow K$是包含图(即$i(x)=x$对于所有$x \in G)$，则序列
$$0 \rightarrow G \stackrel{i}{\longrightarrow} K \stackrel{p}{\longrightarrow} K / G \rightarrow 0$$

$$0 \rightarrow G_i \stackrel{f_i}{\longrightarrow} K_i \stackrel{g_i}{\longrightarrow} H_i \rightarrow 0$$

$$0 \rightarrow \bigoplus_{i \in I} G_i \stackrel{\oplus f_i}{\longrightarrow} \bigoplus_{i \in I} K_i \stackrel{\oplus g_i}{\longrightarrow} \bigoplus_{i \in I} H_i \rightarrow 0$$

## 数学代写|代数拓扑代考Algebraic Topology代考|Free Product and Tensor Product of Groups

1.5.1设$G$和$H$为组(不一定是abel)。它们的自由积表示为$G * H$是满足以下条件的群:如果存在同态$i$和$j$，使得对于任意群$K$给定一对同态$f: G \rightarrow K$和$g: H \rightarrow K$，则存在一个唯一的同态$h: G * H \rightarrow K$，使得图1.2中的图可以交换。

1.5.3设$G=\langle X: R\rangle$和$H=\langle Y: S\rangle$表示组$G$和$H$，其中集合$X$和$Y$是生成器(因此关系$R$和$S$)是不相交的。然后介绍$G * H$是由
$$G * H=\langle X \cup Y: R \cup S\rangle$$

## 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。