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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Cayley’s Theorem and its Applications

In earlier sections, we mentioned the difficulty of describing all groups and their elements in a consistent manner. Cayley’s Theorem offers one approach.
Theorem 1.9.27 (Cayley’s Theorem)
Every finite group $G$ is isomorphic to a subgroup of $S_{n}$ for some $n$.
Before giving the proof of this theorem and illustrating it with examples, we mention the importance for computational methods. Because of Cayley’s Theorem implementing operations and methods for permutation groups (subgroups of some $S_{n}$ ) ultimately encompasses all finite groups. For example, in SAGE, the command G=Dihedralgroup (5) defines the group $G$ as the subgroup of $S_{5}$ expressed as
$$\langle(12345),(25)(34)\rangle$$
In particular, every finite group is isomorphic to one constructed as described in Subsection 1.7.3.

Both Maple (GROUPTHEORY package) and SAGE offer many commands that construct specific groups. For example, both have commands for the $n$th dicyclic group, for the Baby Monster Group, for a projective symplectic group over a finite field, and many more.

Proof (of Theorem 1.9.27). Write the elements of $G$ as $G=\left{g_{1}, g_{2}, \ldots, g_{n}\right}$. Define the function $\psi: G \rightarrow S_{n}$ by $\psi(g)=\sigma$ where
$$g g_{i}=g_{\sigma(i)} \quad \text { for all } i \in{1,2, \ldots, n}$$

## 数学代写|抽象代数作业代写abstract algebra代考|Generators and Relations

In Section 1.1.2, where we introduced the abstract notation for the dihedral group $D_{n}$, we saw that all dihedral symmetries can be obtained as various compositions of $r$ (the rotation by angle $2 \pi / n$ ) and $s$ (the reflection through the $x$-axis). We also know that $r^{n}=1$ because $r$ has order $n$, and that $s^{2}=1$ because $s$ has order 2 . However, we also proved that $r$ and $s$ satisfy $s r=r^{-1} s$.
Since every element in $D_{n}$ can be created from operations on $r$ and $s$, we say that $r$ and $s$ generate $D_{\mathrm{n}}$ and that $r^{n}=1, s^{2}=1$, and $s r=r s^{-1}$ are relations on $r$ and $s$. In group theory, we write
$$D_{n}=\left\langle r, s \mid r^{n}=s^{2}=1, s r=r^{-1} s\right\rangle$$
to express that every element in $D_{n}$ can be obtained by a finite number of repeated operations between $r$ and $s$ and that every algebraic relation between $r$ and $s$ can be deduced from the relations $r^{n}=1, s^{2}=1$, and $s r=r^{-1} s$. The expression (1.9) is the standard presentation of $D_{n}$.

The relation $s r=r^{-1} s$ shows that in any term $s^{k} r^{l}$ it is possible to “move” the $s$ to the left of the term by appropriately changing the power on $r$. Hence, in any word involving the generators $r$ and $s$, it is possible, by appropriate changes on powers, to move all the powers of $s$ to the right and all the powers of $r$ to the left. Hence, every expression in the $r$ and $s$ can be rewritten as $r^{l} s^{k}$ with $0 \leq k \leq 1$ and $0 \leq l \leq n-1$. Though we already knew $\left|D_{n}\right|=2 n$ from geometry, this reasoning shows that there are at most $2 n$ terms in the group given by this presentation.

## 数学代写|抽象代数作业代写abstract algebra代考|Cayley’s Theorem and its Applications

$\langle(12345),(25)(34)\rangle$

Maple (GROUPTHEORY 包) 和 SAGE 都提供了许多构建特定组的命令。例如，两者都有用于 $n$th dicyclic group，对于 Baby Monster Group，对于有限域上的射影辛群，等等。

. 定义函数 $\psi: G \rightarrow S_{n}$ 经过 $\psi(g)=\sigma$ 在哪里
$g g_{i}=g_{\sigma(i)} \quad$ for all $i \in 1,2, \ldots, n$

## 数学代写|抽象代数作业代写abstract algebra代考|Generators and Relations

$$D_{n}=\left\langle r, s \mid r^{n}=s^{2}=1, s r=r^{-1} s\right\rangle$$

## 有限元方法代写

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## 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