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# 数学代写|抽象代数代写Abstract Algebra代考|Math4120 Abelian Groups Have Hamiltonian Paths

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## 数学代写|抽象代数代写Abstract Algebra代考|Abelian Groups Have Hamiltonian Paths

Let $G$ be a finite Abelian group, and let $S$ be any generating set for $G$. Then Cay(S:G) has a Hamiltonian path.

PROOF We use induction on $|S|$. If $|S|=1$, say, $S={a}$, then the digraph is just a circle labeled with $e, a, a^2, \ldots, a^{m-1}$, where $|a|=m$. Obviously, there is a Hamiltonian path for this case. Now assume that $|S|>1$. Choose some $s \in S$. Let $T=S-{s}$ – that is, $T$ is $S$ with $s$ removed-and set $H=\left\langle s_1, s_2, \ldots, s_{n-1}\right\rangle$ where $S=\left{s_1, s_2, \ldots, s_n\right}$ and $s=s_n$. (Notice that $H$ may be equal to $G$.)
Because $|T|<|S|$ and $H$ is a finite Abelian group, the induction hypothesis guarantees that there is a Hamiltonian path $\left(a_1, a_2, \ldots, a_k\right)$ in $\operatorname{Cay}(T: H)$. We will show that
$$\left(a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k, s, \ldots, a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k\right) \text {, }$$
where $a_1, a_2, \ldots, a_k$ occurs $|G| /|H|$ times and $s$ occurs $|G| /|H|-1$ times, is a Hamiltonian path in $\operatorname{Cay}(S: G)$.

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

Cayley digraphs are natural models for interconnection networks in computer designs, and Hamiltonicity is an important property in relation to sorting algorithms on such networks. One particular Cayley digraph that is used to design and analyze interconnection networks of parallel machines is the symmetric group $S_n$ with the set of all transpositions as the generating set. Hamiltonian paths and circuits in Cayley digraphs arise in a variety of group theory contexts. A Hamiltonian path in a Cayley digraph of a group is simply an ordered listing of the group elements without repetition. The vertices of the digraph are the group elements, and the arcs of the path are generators of the group. In 1948, R. A. Rankin used these ideas (although not the terminology) to prove that certain bell-ringing exercises could not be done by the traditional methods employed by bell ringers. In 1981, Hamiltonian paths in Cayley digraphs were used in an algorithm for creating computer graphics of Escher-type repeating patterns in the hyperbolic plane. This program can produce repeating hyperbolic patterns in color from among various infinite classes of symmetry groups. The program has now been improved so that the user may choose from many kinds of color symmetry. The 2003 Mathematics Awareness Month poster featured one such image (see mathaware.org/mam/03). Two Escher drawings and their computer-drawn counterparts are given in Figures 28.9 through 28.12.

## 数学代写|抽象代数代写Abstract Algebra代考|Abelian Groups Have Hamiltonian Paths

$$\left(a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k, s, \ldots, a_1, a_2, \ldots, a_k, s, a_1, a_2, \ldots, a_k\right),$$

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

Cayley 有向图是计算机设计中互连网络的自然模型，而哈密顿性是与此类网络上的排序算法相关的重要属性。一种用于设计和分析并行机互连网络的特定凯莱有向图是对称群

## 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。