## 数学代写|抽象代数作业代写abstract algebra代考|MATH413

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

Given a group $G$, both Maple and SAGE offer methods to define the subgroup $\langle S\rangle$ generated by the subset $S \subseteq G$. Because of their central importance, subgroups of symmetric groups are called permutation groups. In most computer algebra systems, permutation groups have their own constructors and methods.

(We remind the reader that in Maple, ending a line with ‘:’ tells Maple to perform the calculation without displaying the result, whereas ending with ‘;’ does display the result.) This code defines the subgroup $G 1$ of $S_6$ as
$$G 1=\langle(16)(142)(356)\rangle .$$
Then we find the order of $G 1$, determine if it is abelian, define the center of G1, and then list the elements in the center of $G 1$. The difference between the last two lines is that calculating the center of a group defines another group object, but displaying the list of elements is another command. (The reader should notice in this example that though $G 1$ is generated by an element of order 2 and an element of order 3 , it has order 24 .)
The following commands in SAGE implement the same thing.

This code constructs the subgroup
$$\left\langle\left(\begin{array}{ll} 2 & 1 \ 3 & 3 \end{array}\right),\left(\begin{array}{ll} 1 & 3 \ 1 & 1 \end{array}\right)\right\rangle$$
of $\mathrm{GL}_2\left(\mathbb{F}_5\right)$. Interestingly enough, replacing GF (5) with QQ, SAGE’s label for $\mathbb{Q}$, defines $2 \times 2$ matrices with coefficients in $\mathbb{Q}$, making $G$ a subgroup of $\mathrm{GL}_2(\mathbb{Q})$. Then H.order () would return +Infinity.

Both Maple and SAGE offer commands related to the normalizer and centralizer of a set, conjugates of elements, and many other topics.

## 数学代写|抽象代数作业代写abstract algebra代考|Lattice of Subgroups

In order to develop an understanding of the internal structure of a group, listing all the subgroups of a group has some value. However, showing how these subgroups are related to each other carries more information. The lattice of subgroups offers a visual representation of containment among subgroups.
Denote by $\operatorname{Sub}(G)$ the set of all subgroups of the group $G$. Obviously, $\operatorname{Sub}(G) \subseteq \mathcal{P}(G)$. For any two subgroups $H, K \in \operatorname{Sub}(G), H \subseteq K$ if and only if $H \leq K$. Consequently, $(\operatorname{Sub}(G), \leq)$ is a poset, namely the subposet of $(\mathcal{P}(G), \subseteq)$ on the subset $\operatorname{Sub}(G)$.

Proof. We know that $(\mathcal{P}(G), \subseteq)$ is a lattice: the least upper bound of any two subsets $A$ and $B$ is $A \cup B$ and the greatest lower bound is $A \cap B$. By Proposition 1.6.10, for any two $H, K \in \operatorname{Sub}(G)$, we also have $H \cap K \in \operatorname{Sub}(G)$, so $H \cap K$ their greatest lower bound in the poset $(\operatorname{Sub}(G), \leq)$.

In contrast, for $H, K \leq G$, the union $H \cup K$ is generally not a subgroup. By Exercise 1.7.13, $\langle H \cup K\rangle$ is the smallest (by inclusion) subgroup of $G$ that contains both $H$ and $K$, and thus the least upper bound of $H$ and $K$ in $(\operatorname{Sub}(G), \leq)$. Since every pair of subgroups of $G$ has a least upper bound and a greatest lower bound in $(\operatorname{Sub}(G), \leq)$, the poset is a lattice.

The construction given in the above proof for a least upper bound of $H$ and $K$, namely $\langle H \cup K\rangle$ is called the join of $H$ and $K$.

Since $(\operatorname{Sub}(G), \leq)$ is a poset, we can create the Hasse diagram for it. By a common abuse of language, we often say “draw the lattice of $G$ ” for “draw the Hasse diagram of the poset $(\operatorname{Sub}(G), \leq)$.” The lattice of a group shows all subgroups and their containment relationships.

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

（我们提醒读者，在 Maple 中，以’:结尾的行告诉 Maple 执行计算而不显示结果，而以’;’结尾的行会显示 结果。）这段代码定义了子组 $G 1$ 的 $S_6$ 作为
$$G 1=\langle(16)(142)(356)\rangle .$$

SAGE 中的以下命令实现了同样的事情。

Maple 和 SAGE 都提供与集合的归一化器和中心化器、 元素共轭以及许多其他主题相关的命令。

## 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。