### 数学代写|抽象代数作业代写abstract algebra代考|MATH355

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 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 $\left(\mathbb{Z}^*, \times\right)$ 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 $\left(\mathbb{Q}^{>0}, \times\right)$ and $\left(\mathbb{R}^{>0}, \times\right)$ 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.

Example 1.2.7. In Section A.6, we introduced modular arithmetic. Recall that $\mathbb{Z} / n \mathbb{Z}$ represents the set of congruence classes modulo $n$ and that $U(n)$ is the subset of $\mathbb{Z} / n \mathbb{Z}$ of elements with multiplicative inverses. Given any integer $n \geq 2$, both $(\mathbb{Z} / n \mathbb{Z},+)$ and $(U(n), \times)$ are groups. The element $\overline{0}$ is the identity in $\mathbb{Z} / n \mathbb{Z}$ and the element $\overline{1}$ is the identity $U(n)$.

The tables for addition in (A.13) and (A.14) are the Cayley tables for $(\mathbb{Z} / 5 \mathbb{Z},+)$ and $(\mathbb{Z} / 6 \mathbb{Z},+)$. By ignoring the column and row for $\overline{0}$ in the multiplication table in Equation (A.13), we obtain the Cayley table for $(U(5), \times)$.

## 数学代写|抽象代数作业代写abstract algebra代考|Notation for Arbitrary Groups

In group theory, we will regularly discuss the properties of an arbitrary group. In this case, instead of writing the operation as $a * b$, where $*$ represents some unspecified binary operation, it is common to write the generic group operation as $a b$. With this convention of notation, it is also common to indicate the identity in an arbitrary group as 1 instead of $e$. In this chapter, however, we will continue to write $e$ for the arbitrary group identity in order to avoid confusion. Finally, with arbitrary groups, we denote the inverse of an element $a$ as $a^{-1}$.

This shorthand of notation should not surprise us too much. We already developed a similar habit with vector spaces. When discussing an arbitrary vector space, we regularly say, “Let $V$ be a vector space.” So though, in a strict sense, $V$ is only the set of the vector space, we implicitly understand that part of the information of a vector space is the addition of vectors (some operation usually denoted $+$ ) and the scalar multiplication of vectors.

By a similar abuse of language, we often refer, for example, to “the dihedral group $D_n$,” as opposed to “the dihedral group $\left(D_n, \circ\right)$.” Similarly, when we talk about “the group $\mathbb{Z} / n \mathbb{Z}$,” we mean $(\mathbb{Z} / n \mathbb{Z},+)$ because $(\mathbb{Z} / n \mathbb{Z}, \times)$ is not a group. And when we refer to “the group $U(n)$,” we mean the group $(U(n), \times)$. We will explicitly list the pair of set and binary operation if there could be confusion as to which binary operation the group refers. Furthermore, as we already saw with $D_n$, even if a group is equipped with a natural operation, we often just write $a b$ to indicate that operation. Following the analogy with multiplication, in a group $G$, if $a \in G$ and $k$ is a positive integer, by $a^k$ we mean
$$a^k \stackrel{\text { def }}{=} \overbrace{a a \cdots a}^{k \text { times }} .$$
We extend the power notation so that $a^0=e$ and $a^{-k}=\left(a^{-1}\right)^k$, for any positive integer $k$.

Groups that involve addition give an exception to the above habit of notation. In that case, we always write $a+b$ for the operation, $-a$ for the inverse, and, if $k$ is a positive integer,
$$k \cdot a \stackrel{\text { def }}{=} \overbrace{a+a+\cdots+a}^{k \text { times }} .$$
We refer to $k \cdot a$ as a multiple of $a$ instead of as a power. Again, we extend the notation to nonpositive “multiples” just as above with powers.

# 抽象代数代写

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

(A.13) 和 (A.14) 中的加法表是 Cayley 表 $(\mathbb{Z} / 5 \mathbb{Z},+)$ 和 $(\mathbb{Z} / 6 \mathbb{Z},+)$. 通过忽略列和行 $\overline{0}$ 在等式
(A.13）的乘法表中，我们得到屾莱表为 $(U(5), \times)$.

## 数学代写|抽象代数作业代写abstract algebra代考|Notation for Arbitrary Groups

$$a^k \stackrel{\text { def }}{=} \overbrace{a a \cdots a}^{k \text { times }} .$$

$$k \cdot a \stackrel{\text { def }}{=} \overbrace{a+a+\cdots+a}^{k \text { times }} .$$

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