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## 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 $z^{a}, z^{b} \in Z_{n}$,
$$f\left(z^{a} z^{b}\right)=\left(z^{a} z^{b}\right)^{2}=\left(z^{a+b}\right)^{2}=z^{2(a+b)}=z^{2 a+2 b}=z^{2 a} z^{2 b}=f\left(z^{a}\right) f\left(z^{b}\right) . \quad \triangle$$
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|Isomorphisms

We have seen some examples where groups, though presented differently, may actually look strikingly the same. For example $\left(Z_{n}, \tau^{-}\right)$and $(\mathbb{Z} / n \mathbb{Z},+)$ behave identically and likewise for $(\mathbb{Z},+)$ and $(2 \mathbb{Z},+)$, where $2 \mathbb{Z}$ means all even numbers. This raises the questions (1) when should we call two groups the same and (2) what would doing so mean.
Definition 1.9.16
Let $G$ and $H$ be two groups. A function $\varphi: G \rightarrow H$ is called an isomorphism if (1) $\varphi$ is a homomorphism and (2) $\varphi$ is a bijection. If there exists an isomorphism between two groups $G$ and $H$, then we say that $G$ and $H$ are $i$ somorphic and we write $G \cong H$.
When two groups are isomorphic, they are for all intents and purposes of group theory the same. We could have defined an isomorphism as a bijection $\varphi$ such that both $\varphi$ and $\varphi^{-1}$ are both homomorphisms. However, this turns out to be heavier than necessary as the following proposition shows.
Proposition 1.9.17
If $\varphi$ is an isomorphism (as defined in Definition $1.9 .16$ ), then $\varphi^{-1}$ : $H \rightarrow G$ is a homomorphism.
Proof. (Left as an exercise for the reader. See Exercise 1.9.25.)

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

$$\varphi\left(g_{1} g_{2}\right)=\varphi\left(g_{1}\right) \varphi\left(g_{2}\right)$$

$$f\left(z^{a} z^{b}\right)=\left(z^{a} z^{b}\right)^{2}=\left(z^{a+b}\right)^{2}=z^{2(a+b)}=z^{2 a+2 b}=z^{2 a} z^{2 b}=f\left(z^{a}\right) f\left(z^{b}\right) . \quad \triangle$$

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

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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