### 数学代写|抽象代数作业代写abstract algebra代考|Math3020A

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## 数学代写|抽象代数作业代写abstract algebra代考|Group Axioms

As we now jump into group theory with both feet, the reader might not immediately see the value in the definition of a group. The plethora of examples we provide subsequent to the definition will begin to showcase the breadth of applications.
Definition 1.2.1
A group is a pair $(G, )$ where $G$ is a set and $$is a binary operation on G that satisfies the following properties: (1) associativity: (a * b) * c=a *(b * c) for all a, b, c \in G; (2) identity: there exists e \in G such that a * e=e * a=a for all a \in G; (3) inverses: for all a \in G, there exists b \in G such that a * b=b * a=e. By Proposition A.2.16, if any binary operation has an identity, then that identity is unique. Similarly, any element in a group has exactly one inverse element. Proposition 1.2.2 Let (G, *) be a group. Then for all a \in G, there exists a unique inverse element to a. Proof. Let a \in G be arbitrary and suppose that b_{1} and b_{2} satisfy the properties of the inverse axiom for the element a. Then$$
\begin{aligned}
b_{1} &=b_{1} * e & & \text { by identity axiom } \
&=b_{1} *\left(a * b_{2}\right) & & \text { by inverse axiom } \
&=\left(b_{1} * a\right) * b_{2} & & \text { by associativity } \
&=e * b_{2} & & \text { by definition of } b_{1} \
&=b_{2} & & \text { by identity axiom. }
\end{aligned}

Therefore, for all $a \in G$ there exists a unique inverse.
Since every group element has a unique inverse, our notation for inverses can reflect this. We denote the inverse element of $a$ by $a^{-1}$.

## 数学代写|抽象代数作业代写abstract algebra代考|A Few Examples

It is important to develop a robust list of examples of groups that show the breadth and restriction of the group axioms.

Example 1.2.4. The pairs $(\mathbb{Z},+),(\mathbb{Q},+),(\mathbb{R},+)$, and $(\mathbb{C},+)$ are groups. In each case, addition is associative and has 0 as the identity element. For a given element $a$, the additive inverse is $-a$.

Example 1.2.5. The pairs $\left(\mathbb{Q}^{}, \times\right),\left(\mathbb{R}^{}, \times\right)$, and $\left(\mathbb{C}^{}, \times\right)$ are groups. Recall that $A^{}$ mean $A-{0}$ when $A$ is a set that includes 0 . In each group, 1 is the multiplicative identity, and, for a given element $a$, the (multiplicative) inverse is $\frac{1}{a}$. Note that $\left(\mathbb{Z}^{*}, x\right)$ is not a group because it fails the inverse axiom. For example, there is no nonzero integer $b$ such that $2 b=1$.

On the other hand $\left(\mathbb{Q}^{>0}, x\right)$ and $\left(\mathbb{R}^{>0}, x\right)$ are groups. Multiplication is a binary operation on $\mathbb{Q}^{>0}$ and on $\mathbb{R}^{>0}$, and it satisfies all the axioms.

Example 1.2.6. A vector space $V$ is a group under vector addition with $\overrightarrow{0}$ as the identity. The (additive) inverse of a vector $\vec{v}$ is $-\vec{v}$. Note that the scalar multiplication of a vector spaces has no bearing on the group properties of vector addition.

## 数学代写|抽象代数作业代写abstract algebra代考|Group Axioms

(1) 关联性: $(a * b) * c=a *(b * c)$ 对所有人 $a, b, c \in G$;
(2)身份: 存在 $e \in G$ 这样 $a * e=e * a=a$ 对所有人 $a \in G$;
(3) 逆: 对所有 $a \in G$ ， 那里存在 $b \in G$ 这样 $a * b=b * a=e$.

$b_{1}=b_{1} * e \quad$ by identity axiom $\quad=b_{1} *\left(a * b_{2}\right) \quad$ by inverse axiom $=\left(b_{1} * a\right) * b_{2}$

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## MATLAB代写

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