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# 数学代写|抽象代数代写Abstract Algebra代考|Finitely generated abelian groups

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## 数学代写|抽象代数代写Abstract Algebra代考|Finitely generated abelian groups

We now arrive at a foundational theorem in the subject. This theorem provides a comprehensive list of all abelian groups that are finitely generated. Unfortunately, the particular details of the proof require techniques that we haven’t covered, but its power is too great to leave this theorem alone. Instead, we’ll prove aspects of the theorem that will give you some ideas as to why it might be true. We’ll begin with an interesting theorem.

Theorem 8.15. Let $G$ be an abelian group. If $T$ is the set of all elements of $G$ with finite order, then $T$ is a subgroup of $G$.

Definition 8.16. Let $G$ be an abelian group. The subgroup $T$ of all elements of $G$ with finite order is called the torsion subgroup of $G$. If the torsion subgroup of $G$ is the trivial group (that is, the only element of $G$ with finite order is $e$ ), then we say $G$ is torsion free.

Exercise 8.17. Many students mistakenly remember the torsion subgroup as “the set of all elements of finite order of a group.” That would be true, if the group itself were abelian. But nonabelian groups are far more problematic. Show that the two matrices $A=\left[\begin{array}{ll}0 & 1 \ 1 & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & \frac{1}{2} \ 2 & 0\end{array}\right]$ each have finite order, but their product has infinite order. This shows that the set of elements of finite order need not be closed in a nonabelian group.

## 数学代写|抽象代数代写Abstract Algebra代考|The first isomorphism theorem

We’re now in a position to put what we’ve learned in the past three chapters together. Specifically, we learned how to construct quotient groups in Chapter 6; the structure of cyclic groups in Chapter 7; and the nature of finitely generated abelian groups in Chapter 8. What we’d like to know now is how to identify the structure of quotient groups. What we need is a way to tell when a quotient group is isomorphic to a well known group, such as a cyclic group or a finitely generated abelian group. Such a method is our first and most important theorem of the chapter.

Theorem 9.1 (The First Isomorphism Theorem). Let $\phi: G \rightarrow G^{\prime}$ be a homomorphism with kernel $K$. Then the function $\bar{\phi}: G / K \rightarrow \phi(G)$ given by $\bar{\phi}(g K)=\phi(g)$ is a welldefined isomorphism.

Corollary 9.2. Let $\phi: G \rightarrow G^{\prime}$ be a surjective homomorphism with kernel $K$. Then $G^{\prime}$ is isomorphic to $G / K$.

With this theorem, we have the power to make intuition precise. Think of quotient groups as collapsing part of the group together, leaving only part of the group left. When this happens, what structure do we have left after the quotient? We’ll make an educated guess and then use the corollary to the First Isomorphism Theorem to verify our guess! The next example and the first few theorems that follow should help develop this intuition.

Example 9.3. Let’s see an example of how to use the First Isomorphism Theorem on a fact we already know: $\mathbb{Z} /\langle n\rangle \cong \mathbb{Z}_n$. What we should do is find an onto homomorphism $\phi: \mathbb{Z} \rightarrow \mathbb{Z}_n$ whose kernel is specifically $\langle n\rangle$. So, let’s use Theorem 7.7 and define a homomorphism $\phi$ by $\phi(1)=1$ (so that $\phi(x)=\phi(x \cdot 1)=x \phi(1)$, which means that $\phi(x)$ is the remainder of $x$ divided by $n$ ). We simply need to show that $\phi$ is onto and that $\operatorname{Ker}(\phi)=\langle n\rangle$.

The first is easy: for any $b \in \mathbb{Z}_n$, we have $\phi(b)=\phi(b \cdot 1)=b \phi(1)=b$. We now need to compute the kernel of $\phi$, and since $\operatorname{Ker}(\phi)$ are all those integers $x$ such that $\phi(x)=0$, we need to find all integers $x$ such that $\phi(x)$ is a multiple of $n$. But $\phi(x)$ is simply the remainder of $x$ when divided by $n$. That means that $x$ itself must be a multiple of $n$, so $\operatorname{Ker}(\phi)$ is the set of all multiples of $n$. That’s what $\langle n\rangle$ is, and thus $\operatorname{Ker}(\phi)=\langle n\rangle$. By the First Isomorphism Theorem, $\mathbb{Z} /\langle n\rangle \cong \mathbb{Z}_n$.

## 数学代写|抽象代数代写Abstract Algebra代考|Finitely generated abelian groups

8.16.定义设$G$是一个阿贝尔群。所有具有有限阶的$G$元素的子群$T$称为$G$的扭转子群。如果$G$的扭转子群是平凡群(即$G$的唯一有限阶元素是$e$)，那么我们说$G$是无扭转的。

## 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。