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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

## 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 $G$, 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 $\left\langle r^2\right\rangle=\left\langle r^4\right\rangle$.
Example 1.7.4. Consider the group $S_4$. Let $H=\langle(13),(1234)\rangle$. We list out all the elements of $H$. By taking powers of the generators, we know that id, $\quad(13), \quad(1234), \quad(1234)^2=(13)(24), \quad(1234)^3=(1432)$ are all in $H$. By operating among these, $H$ also contains $(13)(1234)=(12)(34), \quad(13)(1234)^2=(24), \quad(13)(1234)^3=(14)(23)$.
It is also easy to calculate that $(1234)(13)=(14)(23),(1234)^2(13)=$ $(13)(24)$, and $(1234)^3(13)=(12)(34)$. At this point, we may suspect that $H={\mathrm{id},(13),(1234),(13)(24),(1432),(24),(12)(34),(14)(23)}$
but we have not yet proven that $H$ does not have any other elements.

## 数学代写|抽象代数作业代写abstract algebra代考|Useful CAS Commands

The GroupTheory package in Maple offers the SubgroupLattice command, which offers various ways to visualize the subgroup lattice of a group.

SAGE offers commands to display the Hasse diagram of any poset. By constructing the subgroup poset $\operatorname{Sub}(G)$, we can use this to create a plot object of the subgroup lattice. Since the console cannot display graphics, we use the following SAGE code in a Jupyter notebook.

In this code, we define $G$ as $D_6$ and then define the set subs, which is a set of all the subgroups of $G$. The third line for $f$ defines the partial order of containment, while the fourth line constructs $P$ which is now the poset $(\operatorname{Sub}(G), \subseteq)$. The user defined variable cardlabel is a dictionary type that to each subgroup in subs associates its order. Finally, the last line plots the corresponding Hasse diagram. To run the code, we click the Run button on the Jupyter notebook. This will create a PNG picture of the group lattice. Admittedly, this method does not place the subgroups at levels corresponding to their cardinality. By changing the cardlabel dictionary variable, we can put different labels on the vertices of the diagram. For example, using stir $(x \cdot$ gens ()$)[1:-1]$ insteat of $x$ order () will lahel each suhgromp vertex by the generators of the subgroup.

# 抽象代数代写

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

. 请注意 $\langle r\rangle$ 是旋转的子群。

《1eft 的分隔符缺失或无法识别

$(13), \quad(1234), \quad(1234)^2=(13)(24), \quad(1234)^3=(1432)$ 都在 $H$. 通过在这些之间进行操作， $H$ 还包含
$(13)(1234)=(12)(34), \quad(13)(1234)^2=(24), \quad(13)(1234)^3=(14)(23)$.

$H=\mathrm{id},(13),(1234),(13)(24),(1432),(24),(12)(34),(14)(23)$

## 数学代写|抽象代数作业代写abstract algebra代考|Useful CAS Commands

Maple 中的 GroupTheory 包提供了 SubgroupLattice 命令，该命令提供了各种可视化组的子组格的方法。
SAGE 提供命令来显示任何poset 的Hasse 图。通过构造子群posetSub $(G)$ ，我们可以使用它来创建子群格子的绘图对象。由于控制台无法显示图形，我们在 Jupyter 笔记本中使用以下 SAGE 代码。

## 有限元方法代写

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## MATLAB代写

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师