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# 数学代写|代数拓扑代考Algebraic Topology代考|Linear Representation of a Group

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## 数学代写|代数拓扑代考Algebraic Topology代考|Linear Representation of a Group

A group representation describes an abstract group in terms of linear operators of vector spaces (see Sect. 1.8). This subsection introduces the concept of group representation.

Definition 1.3.1 Let $G$ be a group and $V$ be a vector space over a field $F$. If GL $(V)$ is the general linear group on $V$, then a representation of $G$ on $V$ is a group homomorphism
$$\psi: G \rightarrow \mathrm{GL}(V)$$
such that $\psi\left(g_1 g_2\right)=\psi\left(g_1\right) \circ \psi\left(g_2\right)$ for all $g_1, g_2 \in G$. The vector space $V$ is called the representation space and dimension of $V$ is called the dimension of the representation. The homomorphism $\psi$ is sometimes called a linear representation of the group $G$.

This subsection introduces the concepts of free groups and relations, which are used in computation of fundamental groups and some other groups. The free groups used in multiplication notation here are not necessarily abelian.

Definition 1.3.2 A subset $X=\left{x_j\right}$ of a group $G$ with identity $e$ is called a free set of generators of $G$ if every element $g \in G-{e}$ is uniquely expressable as
$$g=x_1^{i_1} x_2^{i_2} \ldots x_n^{i_n}$$
where $n$ is a positive integer and $i_k \in \mathbf{Z}$. We assume that $x_j \neq x_{j+1}$ for any $j$ (i.e., no adjacent $x_j$ are equal). If $i_j=1$ for some $i_j$, we write $x_j^1$ as $x_j$. Again, if $i_j=0$ for some $i_j$, the term $x_j^0$ is dropped from the expression of $g$.

Example 1.3.3 The expression $g=a^5 b^{-7} c b^8$ is acceptable but the expression $h=$ $a^5 a^{-7} b^0$ is not acceptable.

## 数学代写|代数拓扑代考Algebraic Topology代考|Betti Number and Structure Theorem for Finite Abelian Group

This subsection states some basic concepts and theorems such as fundamental theorem of finitely generated abelian group, Betti number, and structure theorem for finite abelian group which are very key algebraic results used in algebraic topology
Theorem 1.3.29 (Fundamental theorem of finitely generated abelian groups) Every finitely generated abelian group $G$ (not necessarily free) can be expressed uniquely as
$$G \cong \overbrace{\mathbf{Z} \oplus \mathbf{Z} \oplus \cdots \oplus \mathbf{Z}}^{r \text { summands }} \oplus \mathbf{Z}{n_1} \oplus \mathbf{Z}{n_2} \oplus \cdots \oplus \mathbf{Z}{n_t}$$ for some integers $r, n_1, n_2, \ldots, n_t$ such that (i) $r \geq 0$ and $n_j \geq 2$ for all $j$; and (ii) $n_i \mid n{i+1}$, for $1 \leq i \leq t-1$
Definition 1.3.30 The integer $r$ in Theorem 1.3.29 is called the free rank or Betti number of the group $G$ given by E. Betti (1823-1892) and the integers $n_1, n_2, \ldots, n_t$ are called invariant factors of $G$.
Remark 1.3.31 $\overbrace{\mathbf{Z} \oplus \mathbf{Z} \oplus \cdots \oplus \mathbf{Z}}^{r \text { summands }}$ is a free abelian group of rank $r$.
Theorem 1.3.32 (Structure Theorem for finite abelian groups) Any nonzero finite abelian group $G$ can be expressed uniquely as $G \cong \mathbf{Z}{n_1} \oplus \mathbf{Z}{n_2} \oplus \cdots \oplus \mathbf{Z}{n_t}$ such that $n_i \mid n{i+1}$, for $1 \leq i \leq t-1$

Theorem 1.3.33 Two finite abelian groups are isomorphic if and only if they have the same invariant factors.

## 数学代写|代数拓扑代考Algebraic Topology代考|Linear Representation of a Group

$$\psi: G \rightarrow \mathrm{GL}(V)$$

1.3.2子集 $X=\left{x_j\right}$ 一组的 $G$ 有身份 $e$ 的自由发生器集 $G$ 如果每个元素 $g \in G-{e}$ 唯一可表示为
$$g=x_1^{i_1} x_2^{i_2} \ldots x_n^{i_n}$$

## 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。