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# 数学代写|现代代数代考Modern Algebra代写|MATH3230

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## 数学代写|现代代数代考Modern Algebra代写|Even, Odd Permutations

A permutation that can be expressed as a product of an even number of transpositions is called an even permutation, and a permutation that can be expressed as a product of an odd number of transpositions is called an odd permutation.

The product $f g$ in Example 8 was written as a product of six transpositions and then as a product of four transpositions, and $f g$ is an even permutation.
The factorization of an $r$-cycle $\left(i_1, i_2, \ldots, i_r\right)$ as
$$\left(i_1, i_2, \ldots, i_r\right)=\left(i_1, i_r\right)\left(i_1, i_{r-1}\right) \cdots\left(i_1, i_3\right)\left(i_1, i_2\right)$$
uses $r-1$ transpositions. This shows that an $r$-cycle is an even permutation if $r$ is odd and an odd permutation if $r$ is even. The identity is an even permutation since $e=(1,2)(1,2)$. The product of two even permutations is clearly an even permutation. Since any permutation can be written as a product of disjoint cycles, and since the inverse of an $r$-cycle is an $r$-cycle, the inverse of an even permutation is an even permutation. These remarks show that the set $A_n$ of all even permutations in $S_n$ is a subgroup of $S_n$. It is called the alternating group on $n$ elements.

## 数学代写|现代代数代考Modern Algebra代写|Conjugate Elements

If $a$ and $b$ are elements of the group $G$, the conjugate of $a$ by $b$ is the element $b a b^{-1}$. We say that $c \in G$ is a conjugate of $a$ if and only if $c=b a b^{-1}$ for some $b$ in $G$.

We should point out that this concept is trivial in an abelian group $G$, because $b a b^{-1}=b b^{-1} a=e a=a$ for all $b \in G$.

There is a procedure by which conjugates of elements in a permutation group may be computed with ease. To see how this works, suppose that $f$ and $g$ are permutations on ${1,2, \ldots, n}$ that have been written as products of disjoint cycles, and consider $g f g^{-1}$. If $i_1$ and $i_2$ are integers such that $f\left(i_1\right)=i_2$, then $g f g^{-1}$ maps $g\left(i_1\right)$ to $g\left(i_2\right)$, as the following diagram shows:
$$g\left(i_1\right) \stackrel{g^{-1}}{\longrightarrow} i_1 \stackrel{f}{\longrightarrow} i_2 \stackrel{g}{\longrightarrow} g\left(i_2\right) .$$
This means that if
$$\left(i_1, i_2, \ldots, i_r\right)$$
is one of the disjoint cycles in $f$, then
$$\left(g\left(i_1\right), g\left(i_2\right), \ldots, g\left(i_r\right)\right)$$
is a corresponding cycle in $g f g^{-1}$. Thus, if
$$f=\left(i_1, i_2, \ldots, i_r\right)\left(j_1, j_2, \ldots, j_s\right) \cdots\left(k_1, k_2, \ldots, k_t\right),$$
then
$$g f g^{-1}=\left(g\left(i_1\right), g\left(i_2\right), \ldots, g\left(i_r\right)\right)\left(g\left(j_1\right), \ldots, g\left(j_s\right)\right) \cdots\left(g\left(k_1\right), \ldots, g\left(k_t\right)\right) .$$

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Even, Odd Permutations

$$\left(i_1, i_2, \ldots, i_r\right)=\left(i_1, i_r\right)\left(i_1, i_{r-1}\right) \cdots\left(i_1, i_3\right)\left(i_1, i_2\right)$$

## 数学代写|现代代数代考Modern Algebra代写|Conjugate Elements

$$g\left(i_1\right) \stackrel{g^{-1}}{\longrightarrow} i_1 \stackrel{f}{\longrightarrow} i_2 \stackrel{g}{\longrightarrow} g\left(i_2\right) .$$

$$\left(i_1, i_2, \ldots, i_r\right)$$

$$\left(g\left(i_1\right), g\left(i_2\right), \ldots, g\left(i_r\right)\right)$$

$$f=\left(i_1, i_2, \ldots, i_r\right)\left(j_1, j_2, \ldots, j_s\right) \cdots\left(k_1, k_2, \ldots, k_t\right),$$

$$g f g^{-1}=\left(g\left(i_1\right), g\left(i_2\right), \ldots, g\left(i_r\right)\right)\left(g\left(j_1\right), \ldots, g\left(j_s\right)\right) \cdots\left(g\left(k_1\right), \ldots, g\left(k_t\right)\right) .$$

## 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。