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# 数学代写|现代代数代考Modern Algebra代写|MATG5020 Cayley’s theorem and Cayley graphs

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## 数学代写|现代代数代考Modern Algebra代写|Cayley’s theorem and Cayley graphs

Recall that a permutation of a set $X$ is just a bijection $\rho: X \rightarrow X$ on that set and permutations on $X$ form a group called the symmetric group $S(X)$. When the set is finite, we can write it as ${1,2, \ldots, n}$, and $S_n$ denotes the its symmetric group.
Cayley’s theorem can be stated for infinite groups as well as finite groups.
Theorem 4.27 (Cayley). Let $G$ be a group, and let $S(G)$ be the symmetric group on $G$, that is, the group of permutations on the underlying set of $G$. The function $\varphi: G \rightarrow S(G)$ defined by $\varphi(a)(x)=a x$ is a group monomorphism. Therefore, $G$ is isomorphic to a subgroup of $S(G)$

Proof. $\varphi(a)$ is the permutation on $G$ that maps $x$ to $a x$. It’s a bijection since its inverse sends $x$ to $a^{-1} x$. To show that it’s a group homomorphism, it is only necessary to show that $\varphi(a b)=\varphi(a) \varphi(b)$ for $a$ and $b$ in $G$. But $\varphi(a b)(x)=a b x$, and $(\varphi(a) \varphi(b))(x)=\varphi(a)(\varphi(b)(x))=$ $\varphi(a)(b x)=a b x$. Finally, $\varphi: G \rightarrow S(G)$ is a monomorphism since if $\varphi(a)=\varphi(b)$, then evaluating the two permutations at 1 gives $a 1=b 1$, so $a=b$.

Q.E.D.

## 数学代写|现代代数代考Modern Algebra代写|Some small finite groups

We’ve seen a few families of finite groups including $C_n$ the cyclic group of order $n, D_n$ the dihedral group of order $2 n, S_n$ the symmetric group of order $n$ !, and $A_n$ the alternating group of order $n ! / 2$.

The classification of finite groups (up to isomorphism, of course) is extremely difficult. Daniel Gorenstein (1923-1992) was a leader of mathematicians who eventually classified finite simple groups. He was faculty member at Clark University for 13 years.

We’ll look at a few more small finite groups. Later, we’ll look at the classification of finite Abelian groups, and find that they’re all products of cyclic groups.
Table 4.1 lists the small groups up to isomorphism of order up through 24.
$\begin{array}{lrrrrrrrrrrrrrrr}\text { order } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \ \text { number of groups } & 1 & 1 & 1 & 2 & 1 & 2 & 1 & 5 & 2 & 2 & 1 & 5 & 1 & 2 & 1\end{array}$
We won’t prove that these are all of them, but we will look at them all. There are combinatorial theorems, the most important being the Sylow theorems, that help in classifying finite groups.

# 现代代数代写

QED

## 数学代写|现代代数代考Modern Algebra代写|Some small finite groups

$\begin{array}{llllllllllllllll}\text { order } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \ \text { number of groups }\end{array}$

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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。