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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

英国补考|现代代数代写Modern Algebra代考|Groups

Historically, the concept of a group arose through the study of bijective mappings $A(S)$ on a non-empty set $S$. Any mathematical concept comes naturally in a very concrete form from specific sources. We start with two familiar algebraic systems: $(\mathbf{Z},+)$ (under usual addition of integers) and $\left(\mathbf{Q}^{+}, \cdot\right)$ (under usual multiplication of positive rational numbers). We observe that the system $(\mathbf{Z},+)$ possesses the following properties:

1. ‘ $+$ ‘ is a binary operation on $\mathbf{Z}$;
2. ‘+’ is associative in $\mathbf{Z}$;
3. $\mathbf{Z}$ contains an additive identity element, i.e., there is a special element, namely 0 , such that $x+0=0+x=x$ for every $x$ in $\mathbf{Z}$;
4. $\mathbf{Z}$ contains additive inverses, i.e., to each $x \in \mathbf{Z}$, there is an element $(-x)$ in $\mathbf{Z}$, called its negative, such that $x+(-x)=(-x)+x=0$.
We also observe that the other system $\left(\mathbf{Q}^{+}, \cdot\right)$ possesses similar properties:
5. multiplication ‘ ” is a binary operation on $\mathbf{Q}^{+}$;
6. multiplication ‘ ‘ is associative in $\mathbf{Q}^{+}$;
7. $\mathbf{Q}^{+}$contains a multiplicative identity element, i.e., there is a special element, namely 1 , such that $x \cdot 1=1 \cdot x=x$ for every $x$ in $\mathbf{Q}^{+}$;
8. $\mathbf{Q}^{+}$contains multiplicative inverses, i.e., to each $x \in \mathbf{Q}^{+}$, there is an element $x^{-1}$ which is the reciprocal of $x$ in $\mathbf{Q}^{+}$, such that $x \cdot\left(x^{-1}\right)=\left(x^{-1}\right) \cdot x=1$.

If we consciously ignore the notation and terminology, the above four properties are identical in the algebraic systems $(\mathbf{Z},+)$ and $\left(\mathbf{Q}^{+}, \cdot\right)$. The concept of a group may be considered as a distillation of the common structural forms of $(\mathbf{Z},+)$ and $\left(\mathbf{Q}^{+}, \cdot\right)$ and of many other similar algebraic systems.

Remark The algebraic system $(A(S), \circ)$ of bijective mappings $A(S)$ on a nonempty set $S$ (under usual composition of mappings) satisfies all the above four properties. This group of transformations is not the only system satisfying all the above properties. For example, the non-zero rationals, reals or complex numbers also satisfy above four properties under usual multiplication. So it is convenient to introduce an abstract concept of a group to include these and other examples.

英国补考|现代代数代写Modern Algebra代考|Subgroups and Cyclic Groups

Arbitrary subsets of a group do not generally invite any attention. But subsets forming groups contained in larger groups create interest. For example, the group of even integers with 0 , under usual addition is contained in the larger group of all integers and the group of positive rational numbers under usual multiplication is contained in the larger group of positive real numbers. Such examples suggest the concept of a subgroup, which is very important in the study of group theory. The cyclic subgroup is an important subgroup and is generated by an element $g$ of a group $G$. It is the smallest subgroup of $G$ which contains $g$. In this section we study subgroups and cyclic groups.

Let $(G, \circ)$ be a group and $H$ a non-empty subset of $G$. If for any two elements $a, b \in H$, it is true that $a \circ b \in H$, then we say that $H$ is closed under this group operation of $G$. Suppose now $H$ is closed under the group operation ‘o’ on $G$. Then we can define a binary operation ${ }{\circ}^{H}: H \times H \rightarrow H$ by $a \circ{H} b=a \circ b$ for all $a, b \in H$. This operation ${ }_{\circ} H$ is said to be the restriction of ‘o’ to $H \times H$. We explain this with the help of the following example.

Example 2.4.1 Consider the additive group $(\mathbf{R},+)$ of real numbers. If $\mathbf{Q}^{}$ is the set of all non-zern rational numbers, then $\mathbf{Q}^{}$ is a non-empty subset of $\mathbf{R}$. Now for any two $a, b \in \mathbf{Q}^{}$, we find that $a b$ (under usual product of rational numbers) is an element of $\mathbf{Q}^{}$. But we cannot say that $\mathbf{Q}^{*}$ is closed under the group operation ‘ $+$ ‘ on $\mathbf{R}$. Consider now the set $\mathbf{Z}$ of integers. We can show easily that $\mathbf{Z}$ is closed under the group operation ‘+’ on $\mathbf{R}$.

现代代数代考

英国补考|现代代数代写Modern Algebra代考|Groups

1. ‘ +’ 是一个二元运算Z;
2. ‘+’ 是关联的 $\mathbf{Z}$;
3. Z包含一个加法单位元素，即有一个特殊元素，即 0 ，使得 $x+0=0+x=x$ 对于每个 $x$ 在 $\mathbf{Z}$;
4. Z包含加法逆元，即对每个 $x \in \mathbf{Z}$, 有一个元素 $(-x)$ 在 $\mathbf{Z}$ ，称为它的负数，这样 $x+(-x)=(-x)+x=0$.
我们还观察到另一个系统 $\left(\mathbf{Q}^{+}, \cdot\right)$ 具有相似的性质:
5. 乘法’ ” 是二元运算 $\mathbf{Q}^{+}$;
6. 乘法”在 $\mathbf{Q}^{+}$;
7. $\mathbf{Q}^{+}$包含一个乘法单位元素，即，有一个特殊元素，即 1 ，使得 $x \cdot 1=1 \cdot x=x$ 对于每个 $x$ 在 $\mathbf{Q}^{+}$;
8. $\mathbf{Q}^{+}$包含乘法逆元，即对每个 $x \in \mathbf{Q}^{+}$, 有一个元素 $x^{-1}$ 这是倒数 $x$ 在 $\mathbf{Q}^{+}$, 这样 $x \cdot\left(x^{-1}\right)=\left(x^{-1}\right) \cdot x=1$.
如果我们有意识地忽略符号和术语，上述四个性质在代数系统中是相同的 $(\mathbf{Z},+)$ 和 $\left(\mathbf{Q}^{+}, .\right)$. 组的概念可以被认为是对常见结构形式的提炼 $(\mathbf{Z},+)$ 和 $\left(\mathbf{Q}^{+}, .\right)$和许多 其他类似的代数系统。
备注 代数系统 $(A(S), \circ)$ 双射映射 $A(S)$ 在非空集上 $S$ (在通常的映射组合下) 满足以上四个属性。这组变换并不是唯一满足上述所有属性的系统。例如，非零有 理数、实数或复数在通常的乘法下也满足上述四个性质。因此，引一个组的抽象概念来包含这些和其他示例是很方便的。

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