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# 数学代写|现代代数代考Modern Algebra代写|MTH350 Examples of Groups

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## 数学代写|现代代数代考Modern Algebra代写|Examples of Groups

One of the most important examples of groups are groups of transformations, in particular, permutation groups. Permutations were first studied by L.J. Lagrange (1771) and A. Vandermonde (1771). The deep connections between the properties of permutation groups and those of algebraic equations were pointed out by N.H. Abel (1824) and E. Galois (1830). In particular, E. Galois discovered the role played by normal subgroups in problems of solvability of equations by radicals, he also proved that the alternating groups of order $\geq 5$ are simple, etc. Also, the treatise of C. Jordan (1870) on permutation groups played an important role in the systematization and development of this branch of algebra.

Groups are a powerful tool for studying symmetric objects. The concept of a group allows symmetries of geometrical figures to be characterized. An important example of groups are symmetry groups whose elements are symmetries of geometric figures. Namely, we can associate to any geometrical figure $F$ all its isometries which transform $F$ to itself. If this group is not trivial, the figure $F$ is said to be symmetric, or to have symmetry. It was, in fact, the approach of J.S. Fiedorow for the problem of classification of all possible structures of crystals, which is one of the basic problems in crystallography. The study of crystallographic groups was started by J.S. Fiedorow and continued by A. Schönflies at the end of the 19-th century. They showed that there are only 17 plane crystallographic groups and there are exactly 230 different three-dimensional crystallographic groups. It was the first example of the application of group theory to natural science. It is also interesting to note that the three-dimensional crystallographic groups were found mathematically before these 230 different types of crystals were actually discovered in nature.

In 1870 , while studying the theory of groups, C. Jordan showed its close connections with geometry. He studied the classification of main classes of moving a finite rigid solid in the Euclidean space. C. Jordan listed the basic finite groups of rotations and reflections: Cyclic and dihedral groups, groups of rotations and reflections for a tetrahedron, an octahedron and an icosahedron.
In Chapter 2, we introduced the notion of a permutation and showed that permutations form a group under composition. In the first three sections of this chapter, we study the further properties of permutations. The next two sections are devoted to studying the properties of cyclic groups and the groups of symmetries of some plane geometrical figures. In the last section of this chapter, we consider the main properties and the structure theorems of finite Abelian groups.

## 数学代写|现代代数代考Modern Algebra代写|Cycle Notation and Cycle Decomposition of Permutations

In this section, we introduce the cycle notation for permutations. As was shown in the previous chapter permutations can be written as matrices with 2 rows. We will show that they can be also written in the form of product of cycles. For example, the permutation $\sigma=\left(\begin{array}{lll}1 & 2 & 3 \ 2 & 3 & 1\end{array}\right)$ maps the elements of the set $X={1,2,3}$ cyclicly. It is a cycle of degree 3 and, in cycle notation, it is written as (1 23 ). More precisely, we introduce the following definition.

• Definition 3.1.
A permutation $\sigma \in S_{n}=S(X)$ is called a cyclic permutation or a cycle of length $k$ (or $k$-cycle) if there is a subset $Y=\left{a_{1}, a_{2}, \ldots, a_{k}\right} \subseteq$ $X$ such that
$$\sigma\left(a_{1}\right)=a_{2}, \sigma\left(a_{2}\right)=a_{3}, \ldots, \sigma\left(a_{k-1}\right)=a_{k}, \sigma\left(a_{k}\right)=a_{1},$$
and $\sigma\left(a_{i}\right)=a_{i}$ for all $a_{i} \notin Y$.
In this case, a cycle of length $k$ is denoted by $\left(a_{1} a_{2} \ldots a_{k}\right)$.
By assumption, the identity permutation is a cycle of length 0 .

# 现代代数代写

## 数学代写|现代代数代考Modern Algebra代写|Examples of Groups

1870 年，在研究群论时，C. Jordan 展示了它与几何的密切联系。他研究了在欧几里得空间中移动有限刚体的主要灸别的分类。 C. Jordan 列出了旋转和反射的基本有限群：循环和二面体群、四面体、八面体和二十面体的旋转和反射群。

## 数学代写|现代代数代考Modern Algebra代写|Cycle Notation and Cycle Decomposition of Permutations

• 定义 3.1。
一个排列 $\sigma \in S_{n}=S(X)$ 称为循环置换或长度郈环 $k$ (或者 $k$-cycle) 如果有一个子集 \left 的分隔符缺失或无法识别 $\quad X$ 这样
$$\sigma\left(a_{1}\right)=a_{2}, \sigma\left(a_{2}\right)=a_{3}, \ldots, \sigma\left(a_{k-1}\right)=a_{k}, \sigma\left(a_{k}\right)=a_{1},$$
和 $\sigma\left(a_{i}\right)=a_{i}$ 对所有人 $a_{i} \notin Y$.
在这种情况下，长度为一个循环 $k$ 表示为 $\left(a_{1} a_{2} \ldots a_{k}\right)$.
通过假设, 恒等排列是一个长度为 0 的㿟环。

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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。