数学代写|抽象代数作业代写abstract algebra代考|MATH2701

2023年2月7日

couryes-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra相关的作业也就用不着说。

• 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
couryes™为您提供可以保分的包课服务

数学代写|抽象代数作业代写abstract algebra代考|Definition and Examples

Since a subgroup $H$ of a group $G$ must be nonempty, there exists some $x \in H$. Since $H$ is closed under taking inverses, then $x^{-1} \in H$. Since $H$ is closed under the group operation, then $e=x x^{-1} \in H$. Hence, a subgroup contains the identity element. The property of associativity is inherited from associativity in $G$ and so since $e \in H$ and $H$ is closed under taking inverses, $H$ equipped with the binary operation on $G$ is a group in its own right. (As a point of terminology, it is important to understand that we do not say that “a group is closed under an operation.” Such a statement is circular since a binary operation on $G$ by definition maps any pair of elements in $G$ back into $G$. The terminology of “closed under an operation” is a matter of concern only for strict subsets of $G$.)

Example 1.6.2. With the usual addition operation, $\mathbb{Z} \leq \mathbb{Q} \leq \mathbb{R} \leq \mathbb{C}$. With the multiplication operation we have $\mathbb{Q}^* \leq \mathbb{R}^* \leq \mathbb{C}^$. However, $\mathbb{Z}^$ is not a subgroup of $\mathbb{Q}^$, written $\mathbb{Z}^ \leq \mathbb{Q}^$, because even though $\mathbb{Z}^$ is closed under multiplication, it is not closed under taking multiplicative inverses. For example $2^{-1}=\frac{1}{2} \notin \mathbb{Z}^*$

Example 1.6.3. Any group $G$ always has at least two subgroups, the trivial subgroup ${e}$ and all of $G$.

Example 1.6.4. If $G=D_n$, then $R=\left{\iota, r, r^2, \ldots, r^{n-1}\right}$ is a subgroup. This is the subgroup of rotations. Also for all integers $i$ between 0 and $n-1$, the subsets $H_i=\left{\iota, s r^i\right}$ are subgroups. These subgroups of two elements correspond to reflection about various lines of symmetry.

Example 1.6.5. Let $G=S_n$ and consider the subset of permutations that leave the elements ${m+1, m+2, \ldots n}$ fixed. This is a subgroup of $S_n$ that consists of all elements in $S_m$.

数学代写|抽象代数作业代写abstract algebra代考|Subgroups Generated by a Subset

By virtue of Proposition 1.7.2, $\langle S\rangle$ is called the subgroup generated by $S$. (In the analogy with vector spaces, a subgroup generated by a subset is like the span of a set of elements in a vector space, which is a subspace.)

Example 1.7.3. Let $G=D_6$ be the dihedral group on the hexagon. The subgroup $\langle r\rangle$ consists of all powers of $r$, so is $\langle r\rangle=\left{\iota, r, r^2, r^3, r^4, r^5\right}$. Notice that $\langle r\rangle$ is the subgroup of rotations.

The subgroup $\langle s\rangle={1, s}$ consists of only two elements, reflection across the reference axis of symmetry and the identity transformation.

The subgroup $\left\langle s, r^2\right\rangle$ contains the elements $\iota, s, r^2$, and $r^4$, by virtue of taking powers of elements in $\left{s, r^2\right}$. However, $\left\langle s, r^2\right\rangle$ also contains $s r^2$ and $s r^4$. The defining relation on $s$ and $r$ give $r^a s=s r^{6-a}$. Hence, as we apply this relation, the parity on the power of $r$ does not change. Hence,
$$\left\langle s, r^2\right\rangle=\left{\iota, r^2, r^4, s, s r^2, s r^4\right} .$$
Finally, consider the subgroup $\langle s, s r\rangle$. Obviously this subgroup contains $s$ but it also contains $r=s(s r)$. Hence, $\langle s, s r\rangle=D_6$ because it contains all rotations and all reflections.

For any element $a$ in a group $C$, the subgroup $\langle a\rangle$ is a cyclic subgroup of $G$ whose order is precisely the order $|a|$. It is important to note that distinct sets of generators may give that same subgroup. In the previous example, we noted that $\langle r, s r\rangle=\langle r, s\rangle$. This occurs even with cyclic subgroups. For example, in $D_6$, the rotation subgroup is $\langle r\rangle=\left\langle r^5\right\rangle$. In $D_6$, we also have $\left\langle r^2\right\rangle=\left\langle r^4\right\rangle$.

数学代写|抽象代数作业代写abstract algebra代考|Definition and Examples

Imathbb ${Q}^{\wedge} \backslash \backslash e q \backslash m a t h b b{R}^{\wedge} \backslash l e q \backslash m a t h b b{C}^{\wedge}$. 然 而， $\backslash \operatorname{lmathbb}{Z}^{\wedge}$ 不是的子群 $\backslash m a t h b b{Q}^{\wedge}$ ，写
Imathbb ${Z}^{\wedge} \backslash$ leq $\left.\backslash m a t h b b{Q}^{\wedge}\right}$, 因为即使 $\backslash m a t h b b{Z}^{\wedge}$

$\mathrm{R}=$ Vleft:{iota, $r, r \wedge 2$, lddots, $r \wedge{n-1} \backslash r i g h t}$ 是一个子群。这 是旋转子群。也适用于所有整数 $i$ 介于 0 和 $n-1$, 子集 H_i $=\backslash$ left{iota, s r^ilright $}$ 是子群。这两个元素的子群对 应于关于各种对称线的反射。

数学代写|抽象代数作业代写abstract algebra代考|Subgroups Generated by a Subset

Vangle r rrangle= $=\backslash$ left{{iota, $\left.r, r^{\wedge} 2, r^{\wedge} 3, r \wedge 4, r \wedge 5 \backslash r i g h t\right}$. 请 注意 $\langle r\rangle$ 是旋转子群。

Veft $\backslash a n g l e ~ s, r^{\wedge} 2 \backslash r i g h t \backslash r a n g l e=\backslash l e f t\left{\backslash\right.$ iota, $r^{\wedge} 2, r^{\wedge} 4, s, s r$.

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。