### 数学代写|抽象代数作业代写abstract algebra代考|Development of “specialized” theories of groups

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## 数学代写|抽象代数作业代写abstract algebra代考|Permutation Groups

As noted earlier, Lagrange’s work of 1770 initiated the study of permutations in connection with the study of the solution of equations. It was probably the first clear instance of implicit group-theoretic thinking in mathematics. It led directly to the $\mathrm{~ w o ̄ ̄ i k s ̄ ~ o ̂ f ~ K u ̈ f f i n ̃ i , ~ A ̊ b e ́ l , ~ a ̄ n đ ~ G a ̄ l o ̄ i s ~ đ u ̈}$ to the concept of a permutation group.

Ruffini and Abel proved the unsolvability of the quintic by building on the ideas of Lagrange concerning resolvents. Lagrange showed that a necessary condition for the solvability of the general polynomial equation of degree $n$ is the existence of a resolvent of degree less than $n$. Ruffini and Abel showed that such resolvents do not exist for $n>4$. In the process they developed elements of permutation theory. It was Galois, however, who made the fundamental conceptual advances, and who is considered by many as the founder of (permutation) group theory.

He was familiar with the works of Lagrange, Abel, and Gauss on the solution of polynomial equations. But his aim went well beyond finding a method for solvability of equations. He was concerned with gaining insight into general principles, dissatisfied as he was with the methods of his predecessors: “From the beginning of this century,” he wrote, “computational procedures have become so complicated that any progress by those means has become impossible.”

Galois recognized the separation between “Galois theory”-the correspondence between fields and groups-and its application to the solution of equations, for he wrote that he was presenting “the general principles and just one application” of the theory. “Many of the early commentators on Galois theory failed to recognize this distinction, and this led to an emphasis on applications at the expense of the theory” [19].

Galois was the first to use the term “group” in a technical sense-to him it signified a collection of permutations closed under multiplication: “If one has in the same group the substitutions $S$ and $T$, one is certain to have the substitution $S T$.” He recognized that the most important properties of an algebraic equation were reflected in certain properties of a group uniquely associated with the equation-“the group of the equation.” To describe these properties he invented the fundamental notion of normal subgroup and used it to great effect.

While the issue of resolvent equations preoccupied Lagrange, Ruffini, and Abel, Galois’ basic idea was to bypass them, for the construction of a resolvent required great skill and was not based on a clear methodology. Galois noted instead that the existence of a resolvent was equivalent to the existence of a normal subgroup of prime index in the group of the equation. This insight shifted consideration from the resolvent equation to the group of the equation and its subgroups.
Galois defined the group of an equation as follows:
Let an equation be given, whose $m$ roots are $a, b, c, \ldots .$ There will always be a group of permutations of the letters $a, b, c, \ldots$ which has the following property: (1) that every function of the roots, invariant under the substitutions of that group, is rationally known [i.e., is a rational function of the coefficients and any adjoined quantities]; (2) conversely, that every function of the roots, which can be expressed rationally, is invariant under these substitutions [19].

## 数学代写|抽象代数作业代写abstract algebra代考|Abelian Groups

As noted earlier, the main source for abelian group theory was number theory, beginning with Gauss’ Disquisitiones Arithmeticae. (Note also implicit abelian group theory in Euler’s number-theoretic work [33].) In contrast to permutation theory, grouptheoretic modes of thought in number theory remained implicit until about the last third of the nineteenth century. Until that time no explicit use of the term “group” was made, and there was no link to the contemporary, flourishing theory of permutation groups. We now give a sample of some implicit group-theoretic work in number theory, especially in algebraic number theory.

Algebraic number theory arose in connection with Fermat’s Last Theorem, the insolvability in nonzero integers of $x^{n}+y^{n}=z^{n}$ for $n>2$, Gauss’ theory of binary quadratic forms, and higher reciprocity laws (see Chapter 3.2). Algebraic number fields and their arithmetical properties were the main objects of study. In 1846 Dirichlet studied the units in an algebraic number field and established that (in our terminology) the group of these units is a direct product of a finite cyclic group and a free abelian group of finite rank. At about the same time Kummer introduced his “ideal numbers,” defined an equivalence relation on them, and derived, for cyclotomic fields, certain special properties of the number of equivalence classes, the so-called class number of a cyclotomic field-in our terminology, the order of the ideal class group of the cyclotomic field. Dirichlet had earlier made similar studies of quadratic fields.
In 1869 Schering, a former student of Gauss, investigated the structure of Gauss’ (group of) equivalence classes of binary quadratic forms (see Chapter 3 ). He found certain fundamental classes from which all classes of forms could be obtained by composition. In group-theoretic terms, Schering found a basis for the abelian group of equivalence classes of binary quadratic forms.

## 数学代写|抽象代数作业代写abstract algebra代考|Transformation Groups

As in number theory, so in geometry and analysis, group-theoretic ideas remained implicit until the last third of the nineteenth century. Moreover, Klein’s (and Lie’s) explicit use of groups in geometry influenced the evolution of group theory concep tually rather than technically. It signified a genuine shift in the development of group theory from a preoccupation with permutation groups to the study of groups of transformations. (That is not to suggest, of course, that permutation groups were no longer studied.) This transition was also notable in that it pointed to a turn from finite groups to infinite groups.

Klein noted the connection of his work with permutation groups but also realized the departure he was making. He stated that what Galois theory and his own program have in common is the investigation of “groups of changes,” but added that “to be sure,

the objects the changes apply to are different: there [Galois theory] one deals with a finite number of discrete elements, whereas here one deals with an infinite number of elements of a continuous manifold.”‘ To continue the analogy, Klein noted that just as there is a theory of permutation groups, “we insist on a theory of transformations, a study of groups generated by transformations of a given type.”

Klein shunned the abstract point of view in group theory, and even his technical definition of a (transformation) group is deficient:
Now let there be given a sequence of transformations $A, B, C, \ldots .$ If this sequence has the property that the composite of any two of its transformations yields a transformation that again belongs to the sequence, then the latter will be called a group of transformations [33].
Klein’s work, however, broadened considerably the conception of a group and its applicability in other fields of mathematics. He did much to promote the view that group-theoretic ideas are fundamental in mathematics:
The special subject of group theory extends through all of modern mathematics. As an ordering and classifying principle, it intervenes in the most varied domains.

## 数学代写|抽象代数作业代写abstract algebra代考|Permutation Groups

Ruffini 和 Abel 通过建立拉格朗日关于分解的思想证明了五次方程的不可解性。拉格朗日证明了一般多项式度方程的可解性的必要条件n是否存在度数小于的解析器n. Ruffini 和 Abel 表明此类解决方案不存在n>4. 在这个过程中，他们发展了置换理论的元素。然而，是伽罗瓦在概念上取得了根本性的进步，并且被许多人认为是（置换）群论的创始人。

## 数学代写|抽象代数作业代写abstract algebra代考|Abelian Groups

1869 年，高斯的前学生先灵研究了二元二次形式的高斯（组）等价类的结构（见第 3 章）。他发现了某些基本类别，从这些基本类别中可以通过组合获得所有形式的类别。在群论方面，Schering 为二元二次形式的等价类的阿贝尔群找到了基础。

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