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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|抽象代数作业代写abstract algebra代考|Properties of Cyclic Groups

Recall from Chapter 3 that a group $G$ is called cyclic if there is an element $a$ in $G$ such that $G=\left{a^{n} \mid n \in Z\right}$. Such an element $a$ is called a generator of $G$. In view of the notation introduced in the preceding chapter, we may indicate that $G$ is a cyclic group generated by $a$ by writing $G=\langle a\rangle$.

In this chapter, we examine cyclic groups in detail and determine their important characteristics. We begin with a few examples.

• EXAMPLE 1 The set of integers $Z$ under ordinary addition is cyclic. Both 1 and $-1$ are generators. (Recall that, when the operation is addition, $1^{n}$ is interpreted as
$$\underbrace{1+1+\cdots+1}{n \text { terms }}$$ when $n$ is positive and as $$\underbrace{(-1)+(-1)+\cdots+(-1)}{|n| \text { terms }}$$
when $n$ is negative.)

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

The next theorem tells us how many subgroups a finite cyclic group has and how to find them.

Before we prove this theorem, let’s see what it means. Understanding what a theorem means is a prerequisite to understanding its proof. Suppose $G=\langle a\rangle$ and $G$ has order 30 . The first and second parts of the theorem say that if $H$ is any subgroup of $G$, then $H$ has the form $\left\langle a^{30 / k}\right\rangle$ for some $k$ that is a divisor of 30 . The third part of the theorem says that $G$ has one subgroup of each of the orders $1,2,3,5,6,10,15$, and 30 – and no others. The proof will also show how to find these subgroups.

PROOF Let $G=\langle a\rangle$ and suppose that $H$ is a subgroup of $G$. We must show that $H$ is cyclic. If it consists of the identity alone, then clearly $H$ is cyclic. So we may assume that $H \neq{e}$. We now claim that $H$ contains an element of the form $a^{t}$, where $t$ is positive. Since $G=\langle a\rangle$, every element of $H$ has the form $a^{t}$; and when $a^{t}$ belongs to $H$ with $t<0$, then $a^{-t}$ belongs to $H$ also and $-t$ is positive. Thus, our claim is verified. Now let $m$ be the least positive integer such that $a^{m} \in H$. By closure, $\left\langle a^{m}\right\rangle \subseteq H$. We next claim that $H=\left\langle a^{m}\right\rangle$. To prove this claim, it suffices to let $b$ be an arbitrary member of $H$ and show that $b$ is in $\left\langle a^{m}\right\rangle$. Since $b \in G=$ $\langle a\rangle$, we have $b=a^{k}$ for some $k$. Now, apply the division algorithm to $k$ and $m$ to obtain integers $q$ and $r$ such that $k=m q+r$ where $0 \leq r<m$. Then $a^{k}=a^{m q+r}=a^{m q} a^{r}$, so that $a^{r}=a^{-m q} a^{k}$. Since $a^{k}=b \in H$ and $a^{-m q}=\left(a^{m}\right)^{-q}$ is in $H$ also, $a^{r} \in H$. But, $m$ is the least positive integer such that $a^{m} \in H$, and $0 \leq r<m$, so $r$ must be 0. Therefore, $b=a^{k}=a^{m q}=\left(a^{m}\right)^{q} \in\left\langle a^{m}\right\rangle$. This proves the assertion of the theorem that every subgroup of a cyclic group is cyclic.

## 数学代写|抽象代数作业代写abstract algebra代考| Properties of Cyclic Groups

• 示例 1 整数集 $Z$ 在普通加法下是循环的。 1 和 $-1$ 是生成器。（回想一下，当操作是加法时， $1^{n}$ 被解释为
$$\underbrace{1+1+\cdots+1} n \text { terms }$$
什么时候 $n$ 为正数，并且
$$\underbrace{(-1)+(-1)+\cdots+(-1)}|n| \text { terms }$$
什么时候 $n$ 为负数。

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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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师