# 数学代写|抽象代数作业代写abstract algebra代考|МАTH413

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## 数学代写|抽象代数作业代写abstract algebra代考|Useful CAS Commands

The GroupTheory package in Maple offers the SubgroupLattice command, which offers various ways to visualize the subgroup lattice of a group.

SAGE offers commands to display the Hasse diagram of any poset. By constructing the subgroup poset $\operatorname{Sub}(G)$, we can use this to create a plot object of the subgroup lattice. Since the console cannot display graphics, we use the following SAGE code in a Jupyter notebook.

In this code, we define $G$ as $D_6$ and then define the set subs, which is a set of all the subgroups of $G$. The third line for $f$ defines the partial order of containment, while the fourth line constructs $P$ which is now the poset $(\operatorname{Sub}(G), \subseteq)$. The user defined variable cardlabel is a dictionary type that to each subgroup in subs associates its order. Finally, the last line plots the corresponding Hasse diagram. To run the code, we click the Run button on the Jupyter notebook. This will create a PNG picture of the group lattice. Admittedly, this method does not place the subgroups at levels corresponding to their cardinality. By changing the cardlabel dictionary variable, we can put different labels on the vertices of the diagram. For example, using $\operatorname{str}(x$.gens ()$)[1:-1]$ instead of $x$.order() will label each subgroup vertex by the generators of the subgroup.

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

The operation inside the function on the left-hand side is an operation in $G$ while the operation on the right-hand side occurs in the group $H$. With abstract group notation, we write (1.8) as
$$\varphi\left(g_1 g_2\right)=\varphi\left(g_1\right) \varphi\left(g_2\right)$$
but must remember that the group operations occur in different groups.
Example 1.9.2. Fix a positive real number $b$ and consider the function $f(x)=b^x$. Power rules state that for all $x, y \in \mathbb{R}, b^{x+y}=b^x b^y$. In the language of group theory, this identity can be restated by saying that the exponential function $f(x)=b^x$ is a homomorphism from $(\mathbb{R},+)$ to $\left(\mathbb{R}^*, \times\right)$.

Example 1.9.3. The function of inclusion $f:(\mathbb{Z},+) \rightarrow(\mathbb{R},+)$ given by $f(x)=x$ is a homomorphism.

Example 1.9.4. The function $f: Z_n \rightarrow Z_n$ given by $f(x)=x^2$ is a homomorphism. Let $z$ be a generator of $Z_n$. Then for all $z^a, z^b \in Z_n$,
$$f\left(z^a z^b\right)=\left(z^a z^b\right)^2=\left(z^{a+b}\right)^2=z^{2(a+b)}=z^{2 a+2 b}=z^{2 a} z^{2 b}=f\left(z^a\right) f\left(z^b\right) . \triangle$$
Example 1.9.5. Consider the direct sum $Z_2 \oplus Z_2$, where each $Z_2$ has generator $z$. Consider the function $\varphi: Q_8 \rightarrow Z_2 \oplus Z_2$ defined by
$$\varphi(\pm 1)=(e, e) \quad \varphi(\pm i)=(z, e) \quad \varphi(\pm j)=(e, z) \quad \varphi(\pm k)=(z, z) .$$
This is a homomorphism but in order to verify it, we must check that $\varphi$ satisfies (1.8) for all 64 products of terms in $Q_8$. However, we can cut down the work. First notice that for all terms $a, b \in{1, i, j, k}$, the products $(\pm a)(\pm b)=\pm(a b)$ with the sign as appropriately defined. The following table shows $\varphi(a b)$ with $a$ in the columns and $b$ in the rows.

## 数学代写|抽象代数作业代写abstract algebra代考| CAS有用命令

. . . . . .

Maple中的GroupTheory包提供了SubgroupLattice命令，该命令提供了多种可视化组的子组格的方法

SAGE提供了显示任何序集的Hasse图的命令。通过构造子组偏序集$\operatorname{Sub}(G)$，我们可以使用它来创建子组格的绘图对象。由于控制台不能显示图形，我们在Jupyter笔记本中使用以下SAGE代码

## 数学代写|抽象代数作业代写abstract algebra代考|同态

.

$$\varphi\left(g_1 g_2\right)=\varphi\left(g_1\right) \varphi\left(g_2\right)$$
，但必须记住，组操作发生在不同的组中。

$f(x)=x$给出的包含函数$f:(\mathbb{Z},+) \rightarrow(\mathbb{R},+)$是同态。

$f(x)=x^2$给出的函数$f: Z_n \rightarrow Z_n$是同态的。让$z$成为$Z_n$的生成器。然后对于所有$z^a, z^b \in Z_n$，
$$f\left(z^a z^b\right)=\left(z^a z^b\right)^2=\left(z^{a+b}\right)^2=z^{2(a+b)}=z^{2 a+2 b}=z^{2 a} z^{2 b}=f\left(z^a\right) f\left(z^b\right) . \triangle$$

$$\varphi(\pm 1)=(e, e) \quad \varphi(\pm i)=(z, e) \quad \varphi(\pm j)=(e, z) \quad \varphi(\pm k)=(z, z) .$$

## 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。

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