# 数学代写|抽象代数作业代写abstract algebra代考|Math310

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## 数学代写|抽象代数作业代写abstract algebra代考|Cayley Graph of a Presentation

The Cayley graph of a presentation of a group is another visual tool to understand the internal structure of a group, particularly a group of small order. The vertices of the Cayley graph are the elements of the group $G$ and the edges are pairs ${x, y}$ if there is a generator $g$ such that $y=g x$. One variant of the Cayley graph colors the edges accordingly to distinguish which generator corresponds to which edge. Yet another variant is a directed graph that places an arrow from $x$ to $y$ if there is a generator $g$ such that $y=g x$.

It is important to note that Cayley graph depends on the set of generators in the presentation. So if $G=\langle S\rangle=\left\langle S^{\prime}\right\rangle$, where $S$ and $S^{\prime}$ are different subsets of $G$, the set of vertices will be the same, corresponding to elements of $G$, but the edges of the graph will be different.

As an example, it is not hard to show that $S_{4}=\langle(123),(1234)\rangle$. Figure $1.11$ shows the Cayley graph for $S_{4}$ using these generators. The double edges correspond to left multiplication by (123) and the single edges to left multiplication by (1234). This Cayley graph has the adjacency structure of the Archimedean solid named a rhombicuboctahedron.

## 数学代写|抽象代数作业代写abstract algebra代考|Groups in Geometry

Groups arise naturally in many areas of mathematics. Geometry in particular offers many examples of groups. The dihedral group, which we introduced in Section $1.1$ as a motivation for groups, comes from geometry. However, there are countless connections between group theory and geometry, ranging from generalizations of dihedral groups (reflection groups, e.g., [20]) to applications in advanced differential geometry and topology. In fact, group theory became so foundational to geometry that in 1872 , Felix Klein proposed the Erlangen program: to classify all geometries using projective geometry and groups of allowed transformations.

In this section, we introduce a few instances in which groups arise in geometry in an elementary way. This section only offers a glimpse into these topics.

In the real plane $\mathbb{R}^{2}$, the Euclidean distance between two points $P=\left(x_{P}, y_{P}\right)$ and $Q=\left(x_{Q}, y_{Q}\right)$ is
$$d(P, Q)=\sqrt{\left(x_{Q}-x_{P}\right)^{2}+\left(y_{Q}-y_{P}\right)^{2}}$$
The plane equipped with this distance function is called the Euclidean plane. More generally, the Euclidean $n$-space is the set $\mathbb{R}^{n}$ equipped with the Euclidean distance function
$$d(\vec{x}, \vec{y})=\sqrt{\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}}$$

## 数学代写|抽象代数作业代写abstract algebra代考|Groups in Geometry

$$d(P, Q)=\sqrt{\left(x_{Q}-x_{P}\right)^{2}+\left(y_{Q}-y_{P}\right)^{2}}$$

$$d(\vec{x}, \vec{y})=\sqrt{\sum_{i=1}^{n}\left(y_{i}-x_{i}\right)^{2}}$$

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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。

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