## 数学代写|现代代数代写Modern Algebra代考|MAT 423

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## 数学代写|现代代数代写Modern Algebra代考|Symmetry Groups of Geometric Figures in Euclidean Plane

The study of symmetry gives the most appealing applications in group theory. While studying symmetry we use geometric reasoning. Symmetry is a common phenomenon in science. In general a symmetry of a geometrical figure is a one-one transformation of its points which preserves distance. Any symmetry of a polygon of $n$ sides in the Euclidean plane is uniquely determined by its effect on its vertices, say ${1,2, \ldots, n}$. The group of symmetries of a regular polygon of $n$ sides is called the dihedral group of degree $n$ denoted by $D_{n}$ which is a subgroup of $S_{n}$ and contains $2 n$ elements. $D_{n}$ is generated by rotation $r$ of the regular polygon of $n$ sides through an angle $2 \pi / n$ radians in its own plane about the origin in anticlockwise direction and certain reflections $s$ satisfying some relations (see Ex. 19 of Exercises-III).
(a) Isosceles triangle. Figure $2.1$ is symmetric about the perpendicular bisector 1D. So the symmetry group consists of identity and reflection about the line 1D. In terms of permutations of vertices this group is
$$\left{\left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 2 & 3 \end{array}\right),\left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 3 & 2 \end{array}\right)\right} \cong S_{2} .$$
(b) Equilateral triangle. In Fig. 2.2, the symmetry consists of three rotations of magnitudes $\frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{6 \pi}{3}=2 \pi$ about the center $G$ denoted by $r_{1}, r_{2}$ and $r_{3}$, respectively, together with three reflections about the perpendicular lines $1 D, 2 E, 3 F$ denoted by $t_{1}, t_{2}$, and $t_{3}$, respectively. So in terms of permutations of vertices the symmetry group is
\begin{aligned} \left{r_{1}\right.&=\left(\begin{array}{lll} 1 & 2 & 3 \ 2 & 3 & 1 \end{array}\right), r_{2}=\left(\begin{array}{lll} 1 & 2 & 3 \ 3 & 1 & 2 \end{array}\right), r_{3}=\left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 2 & 3 \end{array}\right), \ t_{1} &\left.=\left(\begin{array}{lll} 1 & 2 & 3 \ 1 & 3 & 2 \end{array}\right), t_{2}=\left(\begin{array}{lll} 1 & 2 & 3 \ 3 & 2 & 1 \end{array}\right), t_{3}=\left(\begin{array}{lll} 1 & 2 & 3 \ 2 & 1 & 3 \end{array}\right)\right} \cong S_{3} . \end{aligned}

## 数学代写|现代代数代写Modern Algebra代考|Group of Rotations of the Sphere

A sphere with a fixed center $O$ can be brought from a given position into any other position by rotating the sphere about an axis through $O$. Clearly, the rotations about the same axis have the same result iff they differ by a multiple of $2 \pi$. Thus if $r(S)$ denotes the set of all rotations about the same axis, then we call the rotations $r$ and $r^{\prime}$ in $r(S)$ equal or different iff they differ by a multiple of $2 \pi$ or not. Clearly, the result of two successive rotations in $r(S)$ can also be obtained by a single rotation in $r(S)$. It follows that $r(S)$ forms a group. (The identity is a rotation in $r(S)$ through an angle 0 and the inverse of a rotation $r$ in $r(S)$ has the same angle but in the opposite direction of $r$. Thus if any rotation $r$ in $r(S)$ has the angle of rotation $\theta$ about the axis, then the map $f: r(S) \rightarrow S^{1}$ defined by $f(r)=e^{i \theta}$ is a group isomorphism from the group $r(S)$ onto the circle group $S^{1}$ (see Example 2.3.2).

Remark The rotations of $\mathbf{R}^{2}$ or $\mathbf{R}^{3}$ about the origin are the linear operators whose matrices with respect to the natural basis are orthogonal and have determinant 1 (see Chap. 8).

# 现代代数代考

## 数学代写|现代代数代写Modern Algebra代考|Symmetry Groups of Geometric Figures in Euclidean Plane

(a) 等腰三角形。数字 $2.1$ 是关于垂直平分线 $1 \mathrm{D}$ 对称的。所以对称群由关于一维线的恒等 和反射组成。就顶点的排列而言，该组是
(b) 等边三角形。在图 $2.2$ 中，对称性由三个大小的旋转组成 $\frac{2 \pi}{3}, \frac{4 \pi}{3}, \frac{6 \pi}{3}=2 \pi$ 关于中心 $G$ 表示为 $r_{1}, r_{2}$ 和 $r_{3}$ ，分别与垂直线的三个反射一起 $1 D, 2 E, 3 F$ 表示为 $t_{1}, t_{2}$ ，和 $t_{3}$ ， 分别。所以就顶点的排列而言，对称群是

## 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。