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

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

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


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

群论中的抽象观点慢慢出现。从 1770 年拉格朗日的隐含群论工作开始,抽象群概念的演变花了一百多年的时间。ET Bell 在这个向抽象化和公理化的演变过程中发现了几个阶段:

尽管有些过于简单化了,所有这些概括都倾向于如此,但这仍然是一个有用的框架。确实,在群论的情况下,首先出现的是“孤立现象”——例如排列、二元二次形式、单位根;然后是对“共同特征”的认识——有限群的概念,包括置换群和有限阿贝尔群(参见上面引用的 Frobenius 和 Stickelberger 的论文);接下来搜索“其他实例”——在我们的案例转换组中;最后是“公设”的表述——在这种情况下是群的公设,包括有限和无限的情况。我们现在考虑抽象的中间和最后阶段何时以及如何发生。
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:
这些符号通常不是可转换的 [commutative] 而是关联的……因此,如果整个组乘以任何一个符号,作为更远或更近的因子 [即,在左边或在右边],效果只是重现组 [33]。

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

抽象群体概念在1880 s和1890 s, 尽管在置换和变换领域仍然出现了大量的论文

(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。
前一种情况的一个有趣的例子是 Frobenius 在抽象环境中对 Sylow 定理的验证,Sylow 在 1872 年证明了置换群。这是在 1887 年的一篇题为“西洛定理的新证明”的论文中完成的。尽管 Frobennius 承认每个有限群都可以由一组置换表示这一事实证明了 Sylow 定理必须适用于所有有限群,但他仍然希望抽象地建立该定理:
由于引入到所有这些证明中的对称群与 Sylow 定理的上下文完全不同,我试图找到它的新推导。
有关与 Sylow 定理有关的群论中抽象演变的案例研究,请参见 [28] 和 [32]。

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

群论从几个不同的来源演变而来,产生了各种具体的理论。这些理论是独立发展的,有的从 1770 年开始发展了一百多年,直到 1880 年代初期才在抽象群概念中收敛。抽象群论出现并在接下来的三十到四十年得到巩固。在那个时期的末期(大约 1920 年),人们可以看出群论分化为几个不同的“理论”。以下是群论中一些进步和新方向的最简单的迹象,从1920 s,以及一些主要贡献者的名字和大致日期:
(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]。

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