### 数学代写|抽象代数作业代写abstract algebra代考|Emergence of abstraction in group theory

statistics-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|抽象代数作业代写abstract algebra代考|Emergence of abstraction in group theory

The abstract point of view in group theory emerged slowly. It took over one hundred years from the time of Lagrange’s implicit group-theoretic work of 1770 for the abstract group concept to evolve. E. T. Bell discerns several stages in this process of evolution towards abstraction and axiomatization:
The entire development required about a century. Its progress is typical of the evolution of any major mathematical discipline of the recent period; first, the discovery of isolated phenomena; then the recognition of certain features common to all; next the search for further instances, their detailed calculation and classification; then the emergence of general principles making further calculations, unless needed for some definite application, superfluous; and last, the formulation of postulates crystallizing in abstract form the structure of the system investigated [2].

Although somewhat oversimplified, as all such generalizations tend to be, this is nevertheless a useful framework. Indeed, in the case of group theory, first came the “isolated phenomena”-for example, permutations, binary quadratic forms, roots of unity; then the recognition of “common features”-the concept of a finite group, encompassing both permutation groups and finite abelian groups (cf. the paper of Frobenius and Stickelberger cited above); next the search for “other instances”-in our case transformation groups; and finally the formulation of “postulates”-in this case the postulates of a group, encompassing both the finite and infinite cases. We now consider when and how the intermediate and final stages of abstraction occurred.
In 1854 Cayley gave the first abstract definition of a finite group in a paper entitled “On the theory of groups, as depending on the symbolic equation $\theta^{n}=1 . “$ (In 1858 Dedekind, in lectures on Galois theory at Göttingen, gave another. See 8.2.) Here is Cayley’s definition:
A set of symbols $1, \alpha, \beta, \ldots$, all of them different, and such that the product of any two of them (no matter in what order), or the product of any one of them into itself, belongs to the set, is said to be a group.
Cayley went on to say that:
These symbols are not in general convertible [commutative] but are associative … and it follows that if the entire group is multiplied by any one of the symbols, either as further or nearer factor [i.e., on the left or on the right], the effect is simply to reproduce the group [33].
He then presented several examples of groups, such as the quaternions (under addition), invertible matrices (under multiplication), permutations, Gauss’ quadratic forms, and groups arising in elliptic function theory. Next he showed that every abstract group is (in our terminology) isomorphic to a permutation group, a result now known as Cayley’s theorem.

## 数学代写|抽象代数作业代写abstract algebra代考|Consolidation of the abstract group concept

The abstract group concept spread rapidly during the $1880 \mathrm{~s}$ and $1890 \mathrm{~s}$, although there still appeared a great many papers in the areas of permutation and transformation

groups. The abstract viewpoint was manifested in two ways:
(a) Concepts and results introduced and proved in the setting of “concrete” groups were now reformulated and reproved in an abstract setting;
(b) Studies originating in, and based on, an abstract setting began to appear.
An interesting example of the former case is the reproving by Frobenius, in an abstract setting, of Sylow’s theorem, which was proved by Sylow in 1872 for permutation groups. This was done in 1887, in a paper entitled “A new proof of Sylow’s theorem.” Although Frobennius admitted that the fact that every finite group can be repreesented by a group of permutations proves that Sylow’s theorem must hold for all finite groups, he nevertheless wished to establish the theorem abstractly:
Since the symmetric group, which is introduced into all these proofs, is totally alien to the context of Sylow’s theorem, I have tried to find a new derivation of it.
For a case study of the evolution of abstraction in group theory in connection with Sylow’s theorem see [28] and [32].

Hölder was an important contributor to abstract group theory, and was responsible for introducing a number of group-theoretic concepts abstractly. For example, in 1889 he defined the abstract notion of a quotient group. The quotient group was first seen as the group of the “auxiliary equation,” later as a homomorphic image, and only in Hölder’s time as a group of cosets. He then “completed” the proof of the JordanHölder theorem, namely that the quotient groups in a composition series are invariant up to isomorphism (see Jordan’s contribution, p. 25). For a history of the concept of quotient group see [36].

In 1893 , in a paper on groups of order $p^{3}, p q^{2}, p q r$, and $p^{4}$, Holder introduced the concept of an automorphism of a group abstractly. He was also the first to study simple groups abstractly. (Previously they were considered in concrete cases-as permutation groups, transformation groups, and so on.) As he said: “It would be of the greatest interest if a survey of all simple groups with a finite number of operations could be known.” (By “operations” Hölder meant “elements.”) He then went on to determine the simple groups of order up to 200 .

## 数学代写|抽象代数作业代写abstract algebra代考|Divergence of developments in group theory

Group theory evolved from several different sources, giving rise to various concrete theories. These theories developed independently, some for over one hundred years, beginning in 1770 , before they converged in the early 1880 s within the abstract group concept. Abstract group theory emerged and was consolidated in the next thirty to forty years. At the end of that period (around 1920) one can discern the divergence of group theory into several distinct “theories.” Here is the barest indication of some of these advances and new directions in group theory, beginning in the $1920 \mathrm{~s}$, with the names of some of the major contributors and approximate dates:
(a) Finite group theory. The major problem here, already formulated by Cayley in the 1870 s and studied by Jordan and Hölder. was to find all finite groups of a given order. The problem proved too difficult and mathematicians turned to special cases, suggested especially by Galois theory: to find all simple or all solvable groups (cf. the Feit-Thompson theorem of 1963 , and the classification of all finite simple groups in 1981). See [14], [15], [30].

## 数学代写|抽象代数作业代写abstract algebra代考|Emergence of abstraction in group theory

1854 年，凯莱在一篇题为“论群论，取决于符号方程”的论文中给出了有限群的第一个抽象定义。θn=1.“（1858 年，戴德金在哥廷根的伽罗瓦理论讲座中给出了另一个。见 8.2。）这是凯莱的定义：

Cayley 继续说：

## 数学代写|抽象代数作业代写abstract algebra代考|Consolidation of the abstract group concept

(a) 在“具体”群体的环境中引入和证明的概念和结果现在在抽象环境中重新制定和验证；
(b) 起源于并基于抽象背景的研究开始出现。

Hölder 是抽象群论的重要贡献者，并负责抽象地引入许多群论概念。例如，他在 1889 年定义了商群的抽象概念。商群最初被视为“辅助方程”的群，后来被视为同态图像，并且仅在 Hölder 时代被视为陪集群。然后他“完成”了 JordanHölder 定理的证明，即复合级数中的商群在同构之前是不变的（参见 Jordan 的贡献，第 25 页）。有关商群概念的历史，请参见 [36]。

1893年，在一篇关于秩序群的论文中p3,pq2,pqr， 和p4，Holder抽象地引入了群的自同构的概念。他也是第一个抽象地研究单群的人。（以前它们在具体情况下被考虑——如置换群、变换群等等。）正如他所说：“如果能够知道对所有具有有限数量操作的简单群的调查，那将是最令人感兴趣的。 ” （“操作”Hölder 的意思是“元素”。）然后他继续确定高达 200 的简单组。

## 数学代写|抽象代数作业代写abstract algebra代考|Divergence of developments in group theory

(a) 有限群论。这里的主要问题已经由 Cayley 在 1870 年代提出并由 Jordan 和 Hölder 研究。是找到给定顺序的所有有限群。这个问题被证明太难了，数学家转向特殊情况，特别是伽罗瓦理论提出的：找到所有简单的或所有可解的群（参见 1963 年的 Feit-Thompson 定理和 1981 年对所有有限单群的分类）。见[14]、[15]、[30]。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。