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数学代写|现代代数代写Modern Algebra代考|Definition of a Group
The fundamental notions of set, mapping, binary operation, and binary relation were presented in Chapter 1. These notions are essential for the study of an algebraic system. An algebraic structure, or algebraic system, is a nonempty set in which at least one equivalence relation (equality) and one or more binary operations are defined. The simplest structures occur when there is only one binary operation, as is the case with the algebraic system known as a group.
An introduction to the theory of groups is presented in this chapter, and it is appropriate to point out that this is only an introduction. Entire books have been devoted to the theory of groups; the group concept is extremely useful in both pure and applied mathematics.
A group may be defined as follows.
Group
Suppose the binary operation * is defined for elements of the set $G$. Then $G$ is a group with respect to * provided the following four conditions hold:
- $G$ is closed under *. That is, $x \in G$ and $y \in G$ imply that $x * y$ is in $G$.
- * is associative. For all $x, y, z$ in $G, x *(y * z)=(x * y) * z$.
- $G$ has an identity element $e$. There is an $e$ in $G$ such that $x * e=e * x=x$ for all $x \in G$.
- $G$ contains inverses. For each $a \in G$, there exists $b \in G$ such that $a * b=b * a=e$.
The phrase “with respect to *” should be noted. For example, the set $\mathbf{Z}$ of all integers is a group with respect to addition but not with respect to multiplication (it has no inverses for elements other than \pm 1 ). Similarly, the set $G={1,-1}$ is a group with respect to multiplication but not with respect to addition. In most instances, however, only one binary operation is under consideration, and we say simply that ” $G$ is a group.” If the binary operation is unspecified, we adopt the multiplicative notation and use the juxtaposition $x y$ to indicate the result of combining $x$ and $y$. Keep in mind, though, that the binary operation is not necessarily multiplication.
数学代写|现代代数代写Modern Algebra代考|Properties of Group Elements
Several consequences of the definition of a group are recorded in Theorem 3.4.
Parts $\mathbf{a}$ and $\mathbf{b}$ of the next theorem are statements about uniqueness, and they can be proved by the standard type of uniqueness proof: Assume that two such quantities exist, and then prove the two to be equal.
Properties of Group Elements
Let $G$ be a group with respect to a binary operation that is written as multiplication.
a. The identity element $e$ in $G$ is unique.
b. For each $x \in G$, the inverse $x^{-1}$ in $G$ is unique.
c. For each $x \in G,\left(x^{-1}\right)^{-1}=x$.
d. Reverse order law. For any $x$ and $y$ in $G,(x y)^{-1}=y^{-1} x^{-1}$.
e. Cancellation laws. If $a, x$, and $y$ are in $G$, then either of the equations $a x=a y$ or $x a=y a$ implies that $x=y$.
Proof We prove parts $\mathbf{b}$ and $\mathbf{d}$ and leave the others as exercises. To prove part $\mathbf{b}$, let $x \in G$, and suppose that each of $y$ and $z$ is an inverse of $x$. That is,
$$
x y=e=y x \text { and } x z=e=z x .
$$
Then
$$
\begin{aligned}
y & =e y & & \text { since } e \text { is an identity } \
& =(z x) y & & \text { since } z x=e \
& =z(x y) & & \text { by associativity } \
& =z(e) & & \text { since } x y=e \
& =z & & \text { since } e \text { is an identity. }
\end{aligned}
$$
Thus $y=z$, and this justifies the notation $x^{-1}$ as the unique inverse of $x$ in $G$.
We shall use part $\mathbf{b}$ in the proof of part $\mathbf{d}$. Specifically, we shall use the fact that the inverse $(x y)^{-1}$ is unique. This means that in order to show that $y^{-1} x^{-1}=(x y)^{-1}$, we only need to verify that $(x y)\left(y^{-1} x^{-1}\right)=e=\left(y^{-1} x^{-1}\right)(x y)$. These calculations are straightforward:
$$
\left(y^{-1} x^{-1}\right)(x y)=y^{-1}\left(x^{-1} x\right) y=y^{-1} e y=y^{-1} y=e
$$
and
$$
(x y)\left(y^{-1} x^{-1}\right)=x\left(y y^{-1}\right) x^{-1}=x e x^{-1}=x x^{-1}=e .
$$
现代代数代考
数学代写|现代代数代写Modern Algebra代考|Definition of a Group
第一章给出了集合、映射、二元运算和二元关系的基本概念。这些概念对于代数系统的研究是必不可少的。代数结构或代数系统是定义了至少一个等价关系(等式)和一个或多个二元操作的非空集合。最简单的结构出现在只有一个二进制运算的时候，就像被称为群的代数系统一样。
在本章中介绍了群的理论，并适当地指出，这只是一个介绍。整本整本的书都致力于群体理论;群的概念在纯数学和应用数学中都非常有用。
组可以定义如下。
集团
假设为集合$G$中的元素定义了二元操作。则$G$是关于的群，只要满足以下四个条件:
$G$在*下关闭。也就是说，$x \在G$中，$y \在G$中意味着$x * y$在G$中。
*是结合律。对于$G中的所有$x, y, z$， x *(y * z)=(x * y) * z$。
$G$有一个单位元$e$。在$G$中有一个$e$使得$x * e=e * x=x$对于G$中的所有$x \。
$G$包含逆。对于G$中的每一个$a \，在G$中存在$b \使得$a * b=b * a=e$。
应该注意短语“相对于”。例如，所有整数的集合$\mathbf{Z}$是一个关于加法而不是关于乘法的群(除了\pm 1之外，它没有其他元素的逆)。同样，集合$G={1，-1}$是一个关于乘法而不是关于加法的群。然而，在大多数情况下，只考虑一个二进制操作，我们简单地说“$G$是一个组”。如果未指定二进制操作，则采用乘法表示法，并使用并列式$x y$来表示$x$和$y$组合的结果。但是请记住，二进制操作不一定是乘法。
数学代写|现代代数代写Modern Algebra代考|Properties of Group Elements
群定义的几个结果记录在定理3.4中。
下一个定理的$\mathbf{a}$和$\mathbf{b}$部分是关于唯一性的陈述，它们可以用标准类型的唯一性证明来证明:假设存在两个这样的量，然后证明这两个量相等。
群元素的性质
设$G$是一个关于二进制运算的组，它被写成乘法。
a.“$G$”中的“$e$”是唯一的标识元素。
b.对于每个$x \in G$, $G$的倒数$x^{-1}$是唯一的。
c.对于每个$x \in G,\left(x^{-1}\right)^{-1}=x$。
d.逆序定律。有关任何$x$和$y$，请参阅$G,(x y)^{-1}=y^{-1} x^{-1}$。
e.取消法。如果$a, x$和$y$在$G$中，则公式$a x=a y$或$x a=y a$中的任何一个都意味着$x=y$。
我们证明了$\mathbf{b}$和$\mathbf{d}$部分，其余部分作为练习。为了证明部分$\mathbf{b}$，设$x \in G$，并假设$y$和$z$都是$x$的逆。也就是说，
$$
x y=e=y x \text { and } x z=e=z x .
$$
然后
$$
\begin{aligned}
y & =e y & & \text { since } e \text { is an identity } \
& =(z x) y & & \text { since } z x=e \
& =z(x y) & & \text { by associativity } \
& =z(e) & & \text { since } x y=e \
& =z & & \text { since } e \text { is an identity. }
\end{aligned}
$$
因此是$y=z$，这证明了将$x^{-1}$标记为$G$中$x$的唯一逆表示。
我们将在$\mathbf{d}$的证明中使用$\mathbf{b}$部分。具体来说，我们将利用逆$(x y)^{-1}$是唯一的这一事实。这意味着为了证明$y^{-1} x^{-1}=(x y)^{-1}$，我们只需要验证$(x y)\left(y^{-1} x^{-1}\right)=e=\left(y^{-1} x^{-1}\right)(x y)$。这些计算很简单:
$$
\left(y^{-1} x^{-1}\right)(x y)=y^{-1}\left(x^{-1} x\right) y=y^{-1} e y=y^{-1} y=e
$$
和
$$
(x y)\left(y^{-1} x^{-1}\right)=x\left(y y^{-1}\right) x^{-1}=x e x^{-1}=x x^{-1}=e .
$$
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