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

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

Definition 2.6. A group consists of a set $G$ together with a composition law
\begin{aligned} G \times G & \longrightarrow G, \ \left(g_1, g_2\right) & \longmapsto g_1 \cdot g_2, \end{aligned}
satisfying the following axioms:
(a) (Identity Axiom) There is an element $e \in G$ such that
$$e \cdot g=g \cdot e=g \quad \text { for all } g \in G \text {. }$$
The element $e$ is called the identity element of $G$.
(b) (Inverse Axiom) For every $g \in G$ there is an element $h \in G$ such that
$$g \cdot h=h \cdot g=e .$$
The element $h$ is denoted $g^{-1}$ and is called the inverse of $g$.
(c) (Associative Law) For all $g_1, g_2, g_3 \in G$, the associative law holds; i.e.,
$$g_1 \cdot\left(g_2 \cdot g_3\right)=\left(g_1 \cdot g_2\right) \cdot g_3 .$$
(d) (Optional Feature: Commutative Law) If it is further true that
$$g_1 \cdot g_2=g_2 \cdot g_1 \text { for all } g_1, g_2 \in G,$$
then $G$ is said to be commutative or abelian. ${ }^{3,4}$
Remark 2.7. The key attribute of a group is that it includes a “rule” or “operation” or “law” satisfying three (or maybe four) axioms for combining two elements of the group to create a third element. Depending on the context, you may find the group law being called “addition” or “multiplication” or “composition,” but assigning a name to the group law is simply a linguistic convenience, ${ }^5$ and if you prefer, you may make up some other name, say “xzyglpqz,” for the group law in your favorite group. ${ }^6$

Remark 2.8. Just as there are many names used for the group operation, there are also many notations, including for example
$$g_1 \cdot g_2, \quad g_1 g_2, \quad g_1 \circ g_2, \quad g_1+g_2, \quad g_1 \star g_2, \ldots$$
If we need to specify the group operation explicitly, we write the group as a pair, for example as $(G, \cdot)$ or $(G,+)$

There are many basic properties of groups that follow directly from the three group axioms. We list some of them in the next proposition.

## 数学代写|抽象代数作业代写abstract algebra代考|Interesting Examples of Groups

In Section 2.1 we saw a couple of groups. It’s time to expand our repertoire.
Example 2.14 (Groups of Integers and Integers Modulo $m$ ). The set of integers $\mathbb{Z}=$ ${\ldots,-2,-1,0,1,2, \ldots}$ is a group if we use addition as the group law. It is an example of an infinite group, that is, a group having infinitely many elements. On the other hand, if we try to use multiplication as the group law, then $\mathbb{Z}$ is not a group. Do you see why not? The set $\mathbb{Z} / m \mathbb{Z}$ of integers modulo $m$ forms a group with addition as the group law. It is a finite group of order $m$.

Example 2.15 (Additive Groups of Real, Rational, and Complex Numbers). The set of real numbers $\mathbb{R}$ forms a group with addition as the group law. Similarly, the sets of rational numbers $\mathbb{Q}$ and complex numbers $\mathbb{C}$ are groups using addition. These groups are sometimes denoted by $\mathbb{R}^{+}, \mathbb{Q}^{+}$, and $\mathbb{C}^{+}$to stress that the group law is addition.

Example 2.16 (Multiplicatives Group of Real, Rational, and Complex Numbers). The set of non-zero real numbers forms a group with multiplication as the group law, as do the sets of non-zero rational numbers $\mathbb{Q}$ and non-zero complex numbers $\mathbb{C}$. These groups are denoted by $\mathbb{R}^, \mathbb{Q}^$, and $\mathbb{C}^*$. We remark that the sets of positive real numbers and positive rational numbers also form groups using multiplication.

Definition 2.17. A group $G$ is a cyclic group if there is an element $g \in G$ with the property that
$$G=\left{\ldots, g^{-3}, g^{-2}, g^{-1}, e, g, g^2, g^3, \ldots\right}$$
(Here $g^{-k}$ is shorthand for the $k$-fold product $g^{-1} \cdot g^{-1} \cdots g^{-1}$.) The element $g$ is called a generator of $G$, but note that $g^{-1}$ is also a generator, and finite cyclic groups tend to have many possible generators.

Example 2.18 (Cyclic Groups). We have already seen some examples of cyclic groups. The group of integers $(\mathbb{Z},+)$ is an infinite cyclic group; it is generated by 1 . The group $(\mathbb{Z} / m \mathbb{Z},+)$ of integers modulo $m$ is a finite cyclic group of order $m$ that is generated by 1 . However, we note that $\mathbb{Z}$ and $\mathbb{Z} / m \mathbb{Z}$ have other generators. Thus -1 also generates $\mathbb{Z}$, and $\mathbb{Z} / m \mathbb{Z}$ is generated by any element $a \bmod m$ satisfying $\operatorname{gcd}(a, m)=1$; see Exercise 2.10.

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

$$G \times G \longrightarrow G,\left(g_1, g_2\right) \longmapsto g_1 \cdot g_2$$

(a) (恒等式公理) 有一个元素 $e \in G$ 这样
$$e \cdot g=g \cdot e=g \quad \text { for all } g \in G .$$

(b) (逆公理) 对于每个 $g \in G$ 有一个元素 $h \in G$ 这样
$$g \cdot h=h \cdot g=e .$$

(c) (结合律) 对于所有 $g_1, g_2, g_3 \in G$, 结合律成立; IE,
$$g_1 \cdot\left(g_2 \cdot g_3\right)=\left(g_1 \cdot g_2\right) \cdot g_3$$
(d) (可选特征: 交换律) 如果进一步为真 $g_1 \cdot g_2=g_2 \cdot g_1$ for all $g_1, g_2 \in G$

“xzyglpqz”。 6

$$g_1 \cdot g_2, \quad g_1 g_2, \quad g_1 \circ g_2, \quad g_1+g_2, \quad g_1 \star g_2, \ldots$$

## 数学代写|抽象代数作业代写abstract algebra代考|Interesting Examples of Groups

$\ldots,-2,-1,0,1,2, \ldots$. 是一个群，如果我们使用加法 作为群律。它是无限群的一个例子，也就是说，一个群 有无限多的元素。另一方面，如果我们尝试使用乘法作 为群律，那么君不是一个组。你明白为什么不吗? 套装 $\mathbb{Z} / m \mathbb{Z}$ 整数模 $m$ 以加法作为群律形成一个群。是一个有 限阶群 $m$.

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