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# 数学代写|拓扑学代写TOPOLOGY代考|MATH393 Free Groups, Generators, and Relations

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## 数学代写|拓扑学代写TOPOLOGY代考|Free Groups, Generators, and Relations

So far, all of our examples of groups have been pretty concrete: we have seen cyclic groups, which can be described in terms of explicit elements, and we have seen groups such as dihedral groups and symmetric groups, which we have interpreted as the symmetries of certain objects. These are important ways of thinking about groups, and many groups naturally arise in this way. However, sometimes we will want to think of groups in a more abstract way: by describing how certain key elements relate to each other.

Let us see how we could have described the dihedral group $D_4$ in such a manner. The first step will be to find several elements in $D_4$ such that every element can be obtained by multiplying these together; such a set of elements will be called a generating set. One possibility is to take all the elements of $D_4$ as a generating set. This is perfectly valid, but it isn’t very efficient. In order to state how all the elements relate to each other, we would need to write down an entire multiplication table, which has 64 entries.

We do better by choosing two elements, which we call $\rho$ and $\sigma$. We let $\rho$ be rotation by $\pi / 2$ in the counterclockwise direction, and we let $\sigma$ be a reflection about the $y$-axis. Every element of $D_4$ can be written as a product of powers of $\rho$ and powers of $\sigma$, possibly with many repetitions. (Exercise: Why?)

However, just specifying that we can build $D_4$ out of products of $\rho$ and $\sigma$ is not enough: that doesn’t tell us, for example, that $\sigma^2$ is the identity. So, we also need to specify certain identities that these two elements satisfy; in this case, the three relations
\begin{aligned} \sigma^2 & =e, \ \rho^4 & =e, \ \sigma \rho & =\rho^{-1} \sigma \end{aligned}
are enough to specify all the group behavior. (Exercise: Why do these relations hold?)

## 数学代写|拓扑学代写TOPOLOGY代考|Free Groups and Free Abelian Groups

Free Groups and Free Abelian Groups. Some groups have particularly simple presentations, in that we do not need any relations in their presentation. Such groups are called free groups.
Example The group $\mathbb{Z}$ of integers is a free group. Its presentation is
$$\mathbb{Z}=\langle 1 \mid\rangle$$
Note that when we specify a free group, we do not put anything after the vertical bar, because there are no relations. In the case of the integers, we can write everything as $n \times 1$, for some $n$. That completely describes the group: we don’t need any further information to cut it down to the right size. Indeed, any relation must take the form $1 \times n=0$ for some $n$, and that is false in $\mathbb{Z}$ for $n \neq 0$.

## 数学代写|拓扑学代写TOPOLOGY代考|Free Groups,Generators, and Relations

$$\sigma^2=e, \rho^4 \quad=e, \sigma \rho=\rho^{-1} \sigma$$

## 数学代写|拓扑学代写TOPOLOGY代考|Free Groups and Free Abelian Groups

$$\mathbb{Z}=\langle 1 \mid\rangle$$

## 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。