assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|抽象代数作业代写abstract algebra代考|Subgroups Generated by a Subset

Let $S$ be a subset of a group $G$. Since subgroups of $G$ are closed under the operation and under taking inverses, any subgroup of $G$ that contains $S$ must contain all elements obtained by repeated operations or inverses from elements in $S$.
Definition 1.7.1
Let $S$ be a nonempty subset of a group $G$. We define $\langle S\rangle$ as the subset of words made from elements in $S$, that is to say
$$\langle S\rangle=\left{s_{1}^{\alpha_{1}} s_{2}^{\alpha_{2}} \cdots s_{n}^{\alpha_{n}} \mid n \in \mathbb{N}, s_{i} \in S, \alpha_{i} \in \mathbb{Z}\right}$$
Note that the $s_{i}$ are not necessarily distinct.
Proposition 1.7.2
For any nonempty subset $S$ of a group $G,\langle S\rangle \leq G$.
Proof. First of all $\langle S\rangle$ is nonempty since it contains $S$. For any two elements $x=s_{1}^{\alpha_{1}} s_{2}^{\alpha_{2}} \cdots s_{n}^{\alpha_{n}}$ and $y=t_{1}^{\beta_{1}} t_{2}^{\beta_{2}} \cdots t_{m}^{\beta_{m}}$ in $\langle S\rangle$, we have
$$x y^{-1}=s_{1}^{\alpha_{1}} s_{2}^{\alpha_{2}} \cdots s_{n}^{\alpha_{n}} t_{m}^{-\beta_{m}} \cdots t_{2}^{-\beta_{2}} t_{1}^{-\beta_{1}} .$$
This product is again an element of $\langle S\rangle$ so by the One-Step Subgroup Criterion $\langle S\rangle \leq G$

By virtue of Proposition $1.7 .2,\langle S\rangle$ is called the subgroup generated by $S$. (In the analogy with vector spaces, a subgroup generated by a subset is like the span of a set of elements in a vector space, which is a subspace.)

数学代写|抽象代数作业代写abstract algebra代考|Center, Centralizer, Normalizer

Proposition $1.7 .2$ gave us a way to construct subgroups of a group. However, a number of subsets defined in terms of equations also turn out to always be subgroups. Many play central roles in understanding the internal structure of a group so we present a few such subgroups here.

Proof. Note that $1 \in Z(G)$, so $Z(G)$ is nonempty. Let $x, y \in Z(G)$. Then
$$(x y) g=x(y g)=x(g y)=(x g) y=(g x) y=g(x y)$$
so $Z(G)$ is closed under the operation. Let $x \in Z(G)$. By definition $x g=g x$ so $g=x^{-1} g x$ and $g x^{-1}=x^{-1} g$. Thus, $x^{-1} \in Z(G)$ and we conclude that $Z(G)$ is closed under taking inverses.

Note that $Z(G)=G$ if and only if $G$ is abelian. On the other hand, $Z(G)={1}$ means that the identity is the only element that commutes with every other element. Intuitively speaking, $Z(G)$ gives a measure of how far $G$ is from being abelian. The center itself is an abelian subgroup. However, $Z(G)$ is not necessarily the largest abelian subgroup of $G$.

Example 1.7.9. Let $F$ be $\mathbb{Q}, \mathbb{R}, \mathbb{C}$, or $\mathbb{F}{p}$ (where $p$ is prime). In this example, we prove that $$Z\left(G L{n}(F)\right)={a I \mid a \neq 0},$$
where $I$ is the identity matrix in $\mathrm{GL}{n}(F)$. By properties of matrix multiplication, for all matrices $B \in \mathrm{GL}{n}(F)$ we have $B(a I)=a(B I)=a B=(a I) B$. Hence, ${a I \mid a \neq 0} \subseteq Z\left(\mathrm{GL}_{n}(F)\right)$. The difficulty lies is proving the reverse inclusion.

Suppose $1 \leq i, j \leq n$ with $i \neq j$. Let $E_{i j}$ be the $n \times n$ matrix consisting of zeros in all entries except for a 1 in the $(i, j)$ th entry. The matrix $E_{i j}$ is not in $\mathrm{GL}{n}(F)$ but $I+E{i j}$ is, since $\operatorname{det}\left(I+E_{i j}\right)=1$. Since $B I=I B$ for all $B \in \mathrm{GL}{n}(F)$, then $B\left(I+E{i j}\right)=\left(I+E_{i j}\right) B$ if and only if $B E_{i j}=E_{i j} B$. Thus, all $B \in Z\left(\mathrm{GL}{n}(F)\right)$ satisfy the matrix product $B E{i j}=E_{i j} B$.

The matrix product $B E_{1 j}$ is the matrix of zeros everywhere except for its $j$ th column being the first column of $B$. Similarly, $E_{1 j} B$ is the matrix of zeros everywhere except for its first row being the $j$ th row of $B$. (See Exercise 1.7.16.) Thus, for a particular $j \geq 2$, the identity $B E_{1 j}=E_{1 j} B$ implies that
$$b_{j k}= \begin{cases}0 & \text { if } k \neq j \ b_{11} & \text { if } k=j .\end{cases}$$
If $B \in Z\left(\mathrm{GL}{n}(F)\right)$, then $B E{1 j}=E_{1 j} B$ for all pairs $2 \leq j \leq n$. Therefore, all off-diagonal elements of $B$ are zero and $b_{j j}=b_{11}$ for all $j$, i.e., all diagonal elements of $B$ are equal. This establishes $Z\left(\mathrm{GL}_{n}(F)\right) \subseteq{a I \mid a \neq 0}$ and we deduce (1.7).

数学代写|抽象代数作业代写abstract algebra代考|Subgroups Generated by a Subset

lleft 的分隔符缺失或无法识别

$$x y^{-1}=s_{1}^{\alpha_{1}} s_{2}^{\alpha_{2}} \cdots s_{n}^{\alpha_{n}} t_{m}^{-\beta_{m}} \cdots t_{2}^{-\beta_{2}} t_{1}^{-\beta_{1}} .$$

数学代写|抽象代数作业代写abstract algebra代考|Center, Centralizer, Normalizer

$$(x y) g=x(y g)=x(g y)=(x g) y=(g x) y=g(x y)$$

$$Z(G \operatorname{Ln}(F))=a I \mid a \neq 0,$$

$$b_{j k}=\left{0 \quad \text { if } k \neq j b_{11} \quad \text { if } k=j .\right.$$

有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师