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## 数学代写|抽象代数代写Abstract Algebra代考|Basic properties of rings

Believe it or not, even with all we’ve accomplished, we still haven’t developed a theory to solve equations as simple as $\frac{1}{2} x+1=0$. That’s because the expression $\frac{1}{2} x+1$ isn’t just about multiplication or addition: it involves both operations. Group theory is all about the properties of sets with a single binary operation, so group theory won’t provide the means to solve this type of linear equation. That means we need to develop a new algebraic structure that comes with more than one operation.

Definition 12.1. Let $R$ be a set with two binary operations on $R$, called addition and denoted + , and multiplication and denoted $\cdot$ Then $\langle R,+, \cdot\rangle$ is a ring if and only if the following hold:
(1) $\langle R,+\rangle$ is an abelian group.
(2) $\langle R, \cdot\rangle$ is an associative binary structure.
(3) $a \cdot(b+c)=a \cdot b+a \cdot c$ and $(b+c) \cdot a=b \cdot a+c \cdot a$ for all $a, b, c \in R$ (the distributive laws).

Before we proceed with examples, some observations about this definition are in order. First, we do just write $a b$ to mean $a \cdot b$ as we did with groups under multiplication, and we simply say that $R$ is a ring without explicitly mentioning the two operations. Second, notice that multiplication in a ring is very, very unstructured. Multiplication is only closed and associative: there need not be a multiplicative identity, and even if there is, there need not be any multiplicative inverses, and just like groups, multiplication need not be commutative. In particular, that means that we should not expect to be able to solve equations involving multiplication, since we need inverses to be able to “undo” an operation. Third, a ring’s additive structure is really, really nice: an abelian group! Addition commutes, and in fact we can now talk about subtraction in a ring by defining $a-b=a+(-b)$. Because of that, we also introduce a notation for the additive identity element: we use the symbol $\mathbf{0}$ to denote the additive identity.

## 数学代写|抽象代数代写Abstract Algebra代考|Homomorphisms

Just as we did with groups, we need to develop a notion of maps between rings that preserve structure. Now that we have two operations to deal with, those maps need to preserve both structures.

Definition 12.18. Let $R$ and $R^{\prime}$ be rings and $\phi: R \rightarrow R^{\prime}$ be a function. Then $\phi$ is a (ring) homomorphism if $\phi(a+b)=\phi(a)+\phi(b)$ and $\phi(a b)=\phi(a) \phi(b)$ for all $a, b \in R$. A ring homomorphism $\phi$ is an isomorphism if and only if $\phi$ is also a bijection, and a ring automorphism is a ring isomorphism from a ring to itself. Two rings are isomorphic if and only if there is a ring isomorphism from one ring to the other.

Notice that since ring homomorphisms and isomorphisms are, in fact, group homomorphisms and group isomorphisms under addition, all of the results from group theory still apply to the additive structure of a ring. On the other hand, since the multiplicative structure of a ring isn’t a group structure, we don’t get all of the nice multiplicative properties we might hope for. However, if you have a ring isomorphism, then all of the algebraic properties like commutativity, unity, and inverses are preserved, as well as unity and inverses being mapped to unity and inverses. Yet even with ring homomorphisms, we do get a few nice properties similar to our results from groups.
Theorem 12.19. Let $\phi: R \rightarrow R^{\prime}$ be a ring homomorphism.
(1) $\phi\left(a^n\right)=\phi(a)^n$ for all $a \in R$ and $n \in \mathbb{Z}^{+}$.
(2) If $S$ is a subring of $R$, then $\phi(S)$ is a subring of $R^{\prime}$, and if $S$ is commutative, then $\phi(S)$ is also commutative.
(3) If $S^{\prime}$ is a subring of $R^{\prime}$, then $\phi^{-1}\left(S^{\prime}\right)$ is a subring of $R$.

## 数学代写|抽象代数代写Abstract Algebra代考|Basic properties of rings

12.1.定义设$R$是一个集合，在$R$上有两个二进制运算，分别是加法运算，记为+，和乘法运算，记为$\cdot$，那么$\langle R,+, \cdot\rangle$是一个环，当且仅当以下条件成立:
(1) $\langle R,+\rangle$是一个阿贝尔群。
(2) $\langle R, \cdot\rangle$是一个结合二元结构。
(3)所有$a, b, c \in R$(分配律)为$a \cdot(b+c)=a \cdot b+a \cdot c$和$(b+c) \cdot a=b \cdot a+c \cdot a$。

## 数学代写|抽象代数代写Abstract Algebra代考|Homomorphisms

12.18.定义让 $R$ 和 $R^{\prime}$ 他响了铃， $\phi: R \rightarrow R^{\prime}$ 是一个函数。然后 $\phi$ 环是同态的吗 $\phi(a+b)=\phi(a)+\phi(b)$ 和 $\phi(a b)=\phi(a) \phi(b)$ 对所有人 $a, b \in R$． 一个环同态 $\phi$ 同构是否当且仅当 $\phi$ 也是一个双射，一个环自同构是一个环到它自己的环同构。两个环是同构的当且仅当从一个环到另一个环存在环同构。

(1)所有$a \in R$和$n \in \mathbb{Z}^{+}$为$\phi\left(a^n\right)=\phi(a)^n$。
(2)如果$S$是$R$的子带，则$\phi(S)$是$R^{\prime}$的子带;如果$S$是可交换的，则$\phi(S)$也是可交换的。
(3)如果$S^{\prime}$是$R^{\prime}$的子带，那么$\phi^{-1}\left(S^{\prime}\right)$就是$R$的子带。

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Cayley’s theorem

What began as a simple example of permutations from geometry gave rise to an interesting concept: recognizing a given group as a subgroup of a symmetric group. Is that possible for all groups? Not only is it true, but the theorem that answers our question is a famous result in abstract algebra.

Lemma 10.14. Let $G$ be a group. For each $g \in G$, define $\lambda_g: G \rightarrow G$ by $\lambda_g(x)=g x$, and let $\Lambda=\left{\lambda_g \mid g \in G\right}$. Likewise, for each $g \in G$, define $\rho_g: G \rightarrow G$ by $\rho_g(x)=x g$, and let $P=\left{\rho_g \mid g \in G\right}$. Then both $\Lambda$ and $P$ are subgroups of $S_G$.

Theorem 10.15 (Cayley’s Theorem). Let $G$ be a group. Then $G$ is isomorphic to a subgroup of $S_G$. In particular, every finite group of order $n$ is isomorphic to a subgroup of $S_n$

Notice that what Cayley’s theorem says is that every group $G$, no matter how large, is a subgroup of the (rather large) symmetric group $S_G$. You might be able to find a subgroup isomorphic to $G$ in a “smaller” symmetric group, and there might be multiple subgroups of $S_G$ isomorphic to $G$. In fact, if you used the lemma in your proof of Cayley’s theorem, then you showed that there are, in fact, at least two ways to view $G$ as a subgroup of $S_G$ : as the subgroups $\Lambda$ and $P$ of $S_G$.

Exercise 10.16. Let’s apply Cayley’s theorem to some known groups. For each small group $G$ below, find a collection of permutations of the elements of $G$ that correspond to $\Lambda$ and to $P$ as given by Cayley’s theorem.
(1) $G=\mathbb{Z}_3$
(3) $G=\mathbb{Z}_2 \times \mathbb{Z}_2$
(2) $G=\mathbb{Z}_4$
(4) $G=S_3$

## 数学代写|抽象代数代写Abstract Algebra代考|Orbits and cycles

Our ability to recognize the dihedral groups as subgroups of the symmetric groups suggests that we might be able to think of any permutation of a set $A$ as “moving” the elements of $A$ around. Let’s see if we can develop this idea by considering what repeated application of an individual permutation in $S_A$ does to an element of the set $A$.

Definition 11.1. Let $A$ be a set, let $a \in A$, and let $\sigma \in S_A$. Then the orbit of $a$ under $\sigma$ is the $\operatorname{set}\left{\sigma^n(a) \mid n \in \mathbb{Z}\right}$
Theorem 11.2. Let $A$ be a set, and let $\sigma \in S_A$. Then the relation $\sim$ on $A$ defined by $a \sim b$ if and only if $b$ is in the orbit of $a$ under $\sigma$
is an equivalence relation on $A$ whose equivalence classes are the orbits of the elements of A under $\sigma$.

Definition 11.3. Let $A$ be a set. A permutation of $A$ is a cycle if and only if the permutation has at most one orbit with more than one element, and the length of a cycle is the number of elements in its largest orbit. A cycle of length one is the trivial cycle, and a cycle of length two is a transposition. We say two nontrivial cycles are disjoint if their largest orbits are disjoint.

Exercise 11.4. List all the orbits of the given element of $S_6$. Which are cycles? Which are transpositions?
(1) $f(1)=5, f(2)=2, f(3)=1, f(4)=6, f(5)=3, f(6)=4$.
(2) $f(1)=2, f(2)=4, f(3)=1, f(4)=5, f(5)=6, f(6)=3$.
(3) $f(1)=1, f(2)=5, f(3)=3, f(4)=6, f(5)=2, f(6)=4$.
(4) $f(1)=4, f(2)=2, f(3)=3, f(4)=1, f(5)=5, f(6)=6$.

(5) $f(1)=3, f(2)=1, f(3)=6, f(4)=4, f(5)=5, f(6)=2$.
(6) $f(1)=1, f(2)=2, f(3)=3, f(4)=4, f(5)=5, f(6)=6$.

## 数学代写|抽象代数代写Abstract Algebra代考|Cayley’s theorem

(1) $G=\mathbb{Z}_3$
(3) $G=\mathbb{Z}_2 \times \mathbb{Z}_2$
(2) $G=\mathbb{Z}_4$
（4） $G=S_3$

## 数学代写|抽象代数代写Abstract Algebra代考|Orbits and cycles

11.1.定义设$A$为一组，设$a \in A$，设$\sigma \in S_A$。那么$\sigma$下面的$a$轨道就是$\operatorname{set}\left{\sigma^n(a) \mid n \in \mathbb{Z}\right}$

11.3.定义设$A$为集合。$A$的排列是一个循环当且仅当该排列最多有一个包含多个元素的轨道，周期的长度是其最大轨道上的元素数。长度为1的循环是平凡循环，长度为2的循环是转置。我们说两个非平凡循环是不相交的如果它们最大的轨道是不相交的。

(1) $f(1)=5, f(2)=2, f(3)=1, f(4)=6, f(5)=3, f(6)=4$。
(2) $f(1)=2, f(2)=4, f(3)=1, f(4)=5, f(5)=6, f(6)=3$。
(3) $f(1)=1, f(2)=5, f(3)=3, f(4)=6, f(5)=2, f(6)=4$。
(4) $f(1)=4, f(2)=2, f(3)=3, f(4)=1, f(5)=5, f(6)=6$。
(5) $f(1)=3, f(2)=1, f(3)=6, f(4)=4, f(5)=5, f(6)=2$。
(6) $f(1)=1, f(2)=2, f(3)=3, f(4)=4, f(5)=5, f(6)=6$。

## 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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

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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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

## avatest™帮您通过考试

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## 数学代写|抽象代数代写Abstract Algebra代考|Homomorphisms and kernel

To begin our second part of our survey of group theory, consider the following two functions: $\phi: n \mathbb{Z} \rightarrow \mathbb{Z}$ given by $\phi(x)=x$, and the function $\psi: \mathbb{Z} \rightarrow \mathbb{Z}_n$ given by $\psi(x)=r$, where $r$ is the remainder of $x$ divided by $n$. Neither map is an isomorphism, since $\phi$ is not surjective and $\psi$ is not injective. Yet despite this fact, the maps do preserve the group structure of the domain; that is, $\phi(x+y)=\phi(x)+\phi(y)$ and $\psi(x+y)=\psi(x)+\psi(y)$ (this latter fact is tedious to check, but true nonetheless). Hence, while neither map matches the two groups up perfectly, they at least relate the group structures faithfully. Consequently, we’ll define such maps precisely and turn our attention to their properties.

Definition 6.1. Let $G$ and $G^{\prime}$ be groups. A function $\phi: G \rightarrow G^{\prime}$ is a (group) homomorphism from $G$ to $G^{\prime}$ if and only if $\phi(a b)=\phi(a) \phi(b)$ for all $a, b \in G$.

Notice then that an isomorphism is simply a bijective homomorphism. It’s worthwhile to see what properties of isomorphisms still hold even when the homomorphism isn’t bijective.
Theorem 6.2. Let $\phi: G \rightarrow G^{\prime}$ be a group homomorphism and let $H<G$.
(1) The image of the identity of $G$ under $\phi$ is the identity of $G^{\prime}$.
(2) $\phi\left(g^{-1}\right)=\phi(g)^{-1}$ for all $g \in G$.
(3) $\phi\left(g^n\right)=(\phi(g))^n$ for all $g \in G$ and integers $n$.
(4) If $g \in G$ has finite order, then $\phi(g)$ has finite order and is a divisor of the order of $g$.
(5) $\phi(H)<G^{\prime}$
(6) If $H$ is abelian, then $\phi(H)$ is abelian.
(7) If $A \subset H$ generates $H$, then $\phi(A)$ generates $\phi(H)$.

## 数学代写|抽象代数代写Abstract Algebra代考|Normal subgroups

Theorem 6.10 gives us the opportunity to do something truly innovative. Recall that an isomorphism matched elements from two groups in such a way that the operations on the two groups also matched. A homomorphism still matches the operation, but the function need not be bijective, so individual elements aren’t always matched up perfectly. However, we just saw that it’s not the individual elements that are matched up: it’s the cosets of the kernel that are matched with elements of $G^{\prime}$. Does that mean that, somehow, we can put a group operation on cosets? What would that even mean?
Let’s first play with the kernel $K$ of a group homomorphism $\phi: G \rightarrow G^{\prime}$. If we pick two elements $a^{\prime}, b^{\prime} \in G^{\prime}$ and take their inverse images, we’ll get two cosets $a K$ and $b K$, where $\phi(a)=a^{\prime}$ and $\phi(b)=b^{\prime}$. On the other hand, if we take the inverse image of the product $a^{\prime} b^{\prime}$, we’ll get some coset $c K$, where $\phi(c)=a^{\prime} b^{\prime}$. But wait: since $\phi$ is a homomorphism, we know that $\phi(a b)=\phi(a) \phi(b)=a^{\prime} b^{\prime}$. Thus, we can choose $c=a b$. It therefore makes sense that we would want to say something like, “Define the product of cosets by $a K \cdot b K=(a b) K$.” Will this always work?

Let’s begin with a group $G$ and an arbitrary subgroup $H<G$. What we’re going to attempt is to define an operation on the set $G / H$ of left cosets of $H$ in $G$. The previous paragraph gives us what looks like the natural binary operation to use: given two left cosets $a H, b H \in G / H$, define
$$(a H)(b H)=(a b) H$$
But any time we define a function on cosets – and a binary operation is a function – we have to prove that the function is well-defined. This is now our first crucial theorem.
Theorem 6.11. Let $G$ be a group and $H<G$. Then the binary operation on $G / H$ given by $(a H)(b H)=(a b) H$ is well defined if and only if $g H=H g$ for all $g \in G$.

In other words, this operation makes sense – and only makes sense – when the left and right cosets of $H$ in $G$ are the same. These subgroups form the backbone of much of group theory.

Definition 6.12. Let $G$ be a group and $H<G$. The subgroup $H$ is a normal subgroup of $G$ if and only if $g H=H g$ for all $g \in G$. If $H$ is a normal subgroup of $G$, we write $H<G$.

Using this definition, the operation $(a H)(b H)=(a b) H$ is well-defined if and only if $H$ is a normal subgroup of $G$. Let’s now verify that $G / H$ is a group under this operation when $H$ is a normal subgroup of $G$.

## 数学代写|抽象代数代写Abstract Algebra代考|Homomorphisms and kernel

(一)身份形象 $G$ 在下面 $\phi$ 是的身份 $G^{\prime}$.
(2) $\phi\left(g^{-1}\right)=\phi(g)^{-1}$ 对全部 $g \in G$.
(3) $\phi\left(g^n\right)=(\phi(g))^n$ 对全部 $g \in G$ 和整数 $n$.
(4) 如果 $g \in G$ 有有限阶，那么 $\phi(g)$ 具有有限阶并且是阶的除数 $g$.
(5) $\phi(H)<G^{\prime}$
(6) 如果 $H$ 是交换矩阵，那么 $\phi(H)$ 是阿贝尔的。
(7) 如果 $A \subset H$ 产生 $H$ ，然后 $\phi(A)$ 产生 $\phi(H)$.

## 数学代写|抽象代数代写Abstract Algebra代考|Normal subgroups

$$(a H)(b H)=(a b) H$$

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Subgroups and generating sets

As we have seen in Chapter 3 , examples of groups are quite varied: sets of numbers or matrices under addition or multiplication, sets of functions under composition, and even finite sets with a suitably constructed table are groups. However, for many of these examples, it’s not the operation that’s different, but rather only the set that changes.
Example 4.1. Consider the groups $G=\mathbb{R}$ and $H=\mathbb{Q}$. Although they are both groups in their own right, there’s a natural relationship between them: not only is $\mathbb{Q}$ a subset of $\mathbb{R}$, but their operations are identical: addition of elements of $\mathbb{Q}$ is identical to addition of those same elements in $\mathbb{R}$. When this special relationship of being both a subset and agreement on the operation occurs, we tend to think not of two different groups, but of one “large” group inside of which lies the “smaller” group as a subset.

There is a technical issue we must first raise. The binary operation of a group $\langle G, \cdot\rangle$ is a function with domain $G \times G$. Given a subset $H \subset G$, we can restrict the domain of – to $H \times H$ and see if this restricted operation – the “same” one used to define a valid operation on $G$ – yields a group (or a binary operation, at least). Hence, we need a preliminary definition to deal with this technicality.

Definition 4.2. Let $\langle G, \cdot\rangle$ be a binary structure and let $H \subset G$. Then the function $: H \times H \rightarrow G$ given by $(a, b)=a \cdot b$ is called the operation induced by $\cdot$
With this terminology, we can now define the key term precisely.
Definition 4.3. Let $G$ be a group. We say a subset $H \subset G$ is a subgroup of $G$ if and only if $H$ is a group under the operation induced by the operation on $G$. We write $H<G$ to denote that $H$ is a subgroup of $G$.

Given a group $G$, there are always two (not necessarily different) subgroups of $G$ : the group $G$ itself, and the subgroup consisting of the identity element alone. Since we’re often interested in subgroups other than those two, let’s name them for future reference.

## 数学代写|抽象代数代写Abstract Algebra代考|The center of a group

All we’ve done so far is identify what subgroups are, so we might try to use them to help us understand the structure of a group. To begin, let’s see if we can use subgroups to measure how close to abelian a given group is.

Theorem 4.13. Let $G$ be a group. Then the subset $Z(G)={g \in G \mid x g=g x$ for all $x \in G}$ is a subgroup of $G$.

Definition 4.14. Let $G$ be a group. The subgroup $Z(G)={g \in G \mid x g=g x$ for all $x \in G}$ is called the center of $G$.
Corollary 4.15. Let $G$ be a group. Then $G$ is abelian if and only if $Z(G)=G$.
Ah, an actual object that detects if a group is abelian or not. But even if a group $G$ isn’t abelian, that doesn’t mean that the center is the trivial subgroup. After all, there might be only a few elements that don’t commute. Can we use subgroups to see if an individual element commutes well?

Theorem 4.16. Let $G$ be a group and $a \in G$. Then the subset $C(a)={g \in G \mid g a=a g}$ is a subgroup of $G$.

Definition 4.17. Let $G$ be a group and $a \in G$. The subgroup $C(a)={g \in G \mid g a=a g}$ is called the centralizer of $a$ in $G$.

This means that centralizers are objects that tell us how well individual elements commute with others in the group. In fact, you might have anticipated this next theorem.
Theorem 4.18. Let $G$ be a group. Then $Z(G)=\bigcap_{a \in G} C(a)$.
Exercise 4.19. Find the centralizers of the elements $\left[\begin{array}{ll}1 & 0 \ 0 & 2\end{array}\right]$ and $\left[\begin{array}{ll}3 & 0 \ 0 & 3\end{array}\right]$ in the group $G L_2(\mathbb{R})$. Use this to state what elements are in $Z\left(G L_2(\mathbb{R})\right)$, and justify your answer.

## 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。

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## 数学代写|抽象代数代写Abstract Algebra代考|Binary operations

When one begins to think on the variety of equations one tries to solve, it doesn’t take long to realize that each subject has its own objects, upon which are defined a variety of processes that you can use to manipulate them: addition on numbers, matrix multiplication on matrices, etc. But despite the apparent differences between them, they all share one fundamental property: whenever you combine two or more of the objects, you get another object of the same type: the sum of two numbers is a number, the product of two $n \times n$ matrices is another matrix, etc. Hence, that’s where we’ll start our study of abstract algebra proper.

To begin, let’s reflect on what we’re really doing when we “combine” objects. We take two objects from a set and apply some kind of rule or process to this pair of objects, and the result produces an object from that same set. Yet what we’ve described here is nothing more than a function whose domain is ordered pairs of elements from a set and whose range is contained back in that same set. This basic observation is codified in the following definition.

Definition 2.1. Let $G$ be a set. Any function, $$, whose domain is G \times G is a binary operation on G if and only if the range of$$ is a subset of $G$. When $$is a binary operation on G, we say that G is closed under$$. We denote the image of $(a, b)$ under $*$ as $a * b$. We write $\langle G, *\rangle$ to indicate that * is a binary operation on $G$, and we say that $\langle G, *\rangle$ is a binary structure.

Although a binary operation is simply a particular function, it’s probably best to review the terms about functions and how they apply to this definition. If you need, read Section A.2 in Appendix A for a refresher on the relevant terminology.
(1) If $*$ is a binary operation on a set $G$, then the element $a * b$ cannot be undefined for any elements $a, b \in G$. For instance, if we let $G=\mathbb{Q}$ and define $\frac{a}{b} * \frac{c}{d}=\frac{a}{b} / \frac{c}{d}$, then this rule isn’t defined when $c=0$, since division by zero isn’t defined.

(2) Likewise, if $*$ is a binary operation on a set $G$, then the element $a * b$ must be welldefined for each pair of elements $a, b \in G$. For instance, if we let $G=\mathbb{Q}$ and define $\frac{a}{b} * \frac{c}{d}=a+c$, then this rule isn’t well-defined, since $\frac{1}{2}$ and $\frac{2}{4}$ represent the same number, but $\frac{a}{b} * \frac{1}{2}=a+1$, and $\frac{a}{b} * \frac{2}{4}=a+2$, which are always different.

Aside. Students often fail to appreciate the import of this example. The issue of a welldefined function always arises whenever the objects in the set have more than one description or form. The standard way to verify that a binary operation $*$ is well-defined is to take two equivalent forms of the objects in your set and prove that the result does not depend on the form of the objects. Functions dealing with rational numbers frequently fall in this category, since every rational number has infinitely many equivalent fractional forms. Hence, to check that a function $f$ is well-defined in this case, you would need to suppose that $\frac{a}{b}=\frac{c}{d}$ and prove that $f\left(\frac{a}{b}\right)=f\left(\frac{c}{d}\right)$.

## 数学代写|抽象代数代写Abstract Algebra代考|Binary tables

When a set $G$ is small, one way to define a binary operation $*$ on $G$ is to list all of the possible combinations of $a * b$ for elements $a, b \in G$. The most convenient way to do this is to set up a table whose entries indicate the result of applying the binary operation to two elements. Specifically, we create a binary table to define a binary operation on the set $G=\left{a_1, \ldots, a_n\right}$ in the following way:
1) Write the elements of $G$ in a column, then write them in a row at the top in the same order as you did in the column.
2) Fill in the corresponding $n^2$ entries with exactly one element in $G$.
3) Define the element $a_i * a_j$ to be the entry in the $i^{\text {th }}$ row and the $j^{\text {th }}$ column.
It’s also easy to verify that your table gives a rule that is both well-defined and is defined everywhere. After all, as long as every entry is filled with at least one element of $G$, then $a * b$ is defined everywhere; and as long as you don’t put more than one element from $G$ in any entry, then your rule is well-defined.

Exercise 2.9. Let $G={a, b, c}$. Suppose we define a binary operation on $G$ with the following table:
\begin{tabular}{|c||c|c|c|}
\hline$*$ & $a$ & $b$ & $c$ \
\hline \hline$a$ & $a$ & $c$ & $c$ \
\hline$b$ & $b$ & $b$ & $b$ \
\hline$c$ & $a$ & $a$ & $b$ \
\hline
\end{tabular}
(So, for instance, $c * a=a$.)
(1) Compute $a * b, \quad b * a, \quad b * b, \quad(a * c) * b, \quad a *(c * b), \quad c *(c * c)$, and $(c * c) * c$.
(2) Determine if the operation is commutative, associative, neither, or both.

There’s really no theory about binary tables, but there are several observations that are useful to have. First, since you can put any of the $n$ elements of $G$ into any of the $n^2$ entries, there are a total of $n^{n^2}$ possible ways to construct a binary operation on $G$. Second, checking to see if a binary table’s operation is commutative is easy: reflect the table along the “main diagonal” and compare with the original table. If they’re the same, then it’s a commutative operation; otherwise, you’ll have at least one pair of elements $a_i, a_j$ such that $a_i * a_j \neq a_j * a_i$.

Associativity, on the other hand, is never easy to check by looking at the table. That’s because checking associativity deals with using the table sequentially. It also means verifying that $(a * b) * c=a *(b * c)$ for all choices of $a, b, c \in G$, which means you’ve got $n^3$ different pairs of triples to compare. That’s just too tedious to do by hand; a computer is almost a necessity if you need to know if your operation is associative.

## 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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

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## 数学代写|抽象代数代写Abstract Algebra代考|Writing proofs

Reading proofs more effectively will help you to understand them more fully and become better at constructing proofs of your own. But there is no doubt that proof construction is difficult. Just as mathematical reading is not like ordinary reading, mathematical writing is not like ordinary writing. A proof is not like a message to tell a housemate that you have gone to the supermarket. Nor is it like a standard algorithm that you can learn and apply. There are standard proof strategies, and you should look out for those. But they are not that standard-to some extent, each proof is different, which can leave students feeling overwhelmed.

However, specific approaches can help. Some people find it useful to think of proof construction as involving two main processes: a formal part and a problem solving part. The formal part involves using the structure of a theorem to write a ‘frame’ for its proof: writing the premises, leaving a gap, then writing the conclusion. With that done, it is often possible to work forward from the premises and backward from the conclusion by formulating relevant things in terms of definitions or by making standard or obvious deductions. If you let it, a formal approach will shoulder quite a bit of the burden of proving. Indeed, for simple proofs, it might on its own be enough.

For more complex proofs, you also need problem solving to fill in the gap. This requires insight, which can come from reasoning about familiar examples, or from writing down possibly relevant theorems and thinking about whether they usefully apply. A proof will not flow from your pen in a single stream of mathematically correct argument-probably it will involve some false starts and periods of being stuck, and some cleaning up so that the writing makes sense. But writing one requires no magic. To demonstrate what I mean, I will reason through a formal part and a problem-solving part for this theorem.

## 数学代写|抽象代数代写Abstract Algebra代考|Who are you as a student?

Whappy to learn, and willing to put in the hours. The problem you will face is that it is easy to be like that for the first two weeks of a course like Abstract Algebra, but hard to sustain for more than about four. By week eight, you might want to lie on the floor, moan quietly and wish for someone to make it all easy. I can’t make it easy-undergraduate mathematics just isn’t easy. But if you find Abstract Algebra difficult, that is not because you are stupid or incapable. It’s because it is difficult. If this is your first full-on theorems-and-proofs course, it is likely to seem both difficult and alarmingly different from earlier mathematics. This can make students wonder whether they have topped out-whether they cannot cope with mathematics at this level or, more prosaically, whether they just don’t like it.

I would encourage you, though, to avoid making either judgement too soon. Many students transitioning to advanced mathematics have to adjust their expectations in two ways, accepting that they will not understand everything and learning to tolerate longer periods of intellectual discomfort. But most do manage that, and reach a point where they are satisfied with what they have learned. Of course, some then decide that pure mathematics is not their thing and that where possible in future they will avoid it. But better to decide from a position of strength, I think; better to know that you could do more but choose not to. Others experience not only new understanding but real joy in trading the more routine aspects of earlier work for logical reasoning and theory building.

To reach a positive position with minimal pain, I think it helps to reflect on your study trajectory, on the decisions you have made. For instance, you probably chose to study at the most prestigious accessible institution. A natural consequence of this is that the material you are taught will be only just within your intellectual reach. If you wished, you could switch to an easier degree or major, switch to a lower-prestige institution, or drop out of higher education and take a different route into professional life. Some students choose to do those things, and more power to themeveryone should think about how to use their time. But most students don’t. Most, when they reflect, decide that although it might be difficult, they do want to stick with their degree. Reflection and recommitment help, though. If you recognize that you’re doing what you’re doing by choice, it becomes easier to put up with its downsides and keep your eye on the prize.

## 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。