# 数学代写|抽象代数作业代写abstract algebra代考|MATH417

## 数学代写|抽象代数作业代写abstract algebra代考|Cayley Graph of a Presentation

The Cayley graph of a presentation of a group is another visual tool to understand the internal structure of a group, particularly a group of small order. The vertices of the Cayley graph are the elements of the group $G$ and the edges are pairs ${x, y}$ if there is a generator $g$ such that $y=g x$. One variant of the Cayley graph colors the edges accordingly to distinguish which generator corresponds to which edge. Yet another variant is a directed graph that places an arrow from $x$ to $y$ if there is a generator $g$ such that $y=g x$.

It is important to note that Cayley graph depends on the set of generators in the presentation. So if $G=\langle S\rangle=\left\langle S^{\prime}\right\rangle$, where $S$ and $S^{\prime}$ are different subsets of $G$, the set of vertices will be the same, corresponding to elements of $G$, but the edges of the graph will be different.

As an example, it is not hard to show that $S_4=\langle(123),(1234)\rangle$. Figure $1.11$ shows the Cayley graph for $S_4$ using these generators. The double edges correspond to left multiplication by (123) and the single edges to left multiplication by (1234). This Cayley graph has the adjacency structure of the Archimedean solid named a rhombicuboctahedron.

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

For hundreds of years, people have adorned the walls of rooms with repetitive patterns. Borders as a crown to a wall, as a chair rail, as molding to a door, or as a frame to a picture are particularly common artistic and architectural details. Frieze patterns are the patterns of symmetries used in borders.

In the usual group of isometries in the plane, the translations form a subgroup isomorphic to $\mathbb{R}^2$. We sometimes use the description of “discrete” for frieze groups in contrast to “continuous” because there is a translation of least positive displacement.

Example 1.11.10. Consider for example the following pattern and let $G$ be the group of isometries of the plane that preserve the structure of the pattern.

The subgroup of translations of $G$ consists of all translations that are an integer multiple of $2 \overrightarrow{P Q}$. Some other transformations in $G$ include

• reflections through a vertical line $L_1$ through $P$ or any line parallel to $L_1$ displaced by an integer multiple of $\overrightarrow{P Q}$;
• reflection through the horizontal line $L_3=\overleftrightarrow{P Q}$;
• rotations by an angle of $\pi$ about $P, Q$, or any point translated from $P$ by an integer multiple of $\overrightarrow{P Q}$.

It is possible to describe $G$ with a presentation. Let $s_i$ be the reflection through $L_i$, for $i=1,2,3$. We claim that
$$G=\left\langle s_1, s_2, s_3 \mid s_1^2=s_2^2=s_3^2=1,\left(s_1 s_3\right)^2=1,\left(s_2 s_3\right)^2=1\right\rangle .$$
In order to prove the claim, we first should check that $s_1, s_2, s_3$ do indeed generate all of $G$. By Exercise $1.11 .8, s_1 s_3$ corresponds to rotation by $\pi$ about $P$ and $s_2 s_3$ corresponds to rotation by $\pi$ about $Q$. By Exercise 1.11.7 $s_2 s_1$ corresponds to a translation by $2 \overrightarrow{P Q}$. In order to obtain a reflection through another vertical line besides $L_1$ or $L_2$, or a rotation about another point besides $P$ or $Q$, we can translate the strip to center it on $P$ and $Q$, apply the desired transformation $\left(s_1, s_2, s_1 s_3\right.$, or $\left.s_2 s_3\right)$, and then translate back. For example, the rotation by an angle of $\pi$ about $Q_3$, can be described by $\left(s_2 s_1\right)^2 s_2 s_3\left(s_2 s_1\right)^{-2}$. This shows that our choice of generators is sufficient to generate $G$.

# 抽象代数代考

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

$G$的翻译子组由$2 \overrightarrow{P Q}$整数倍的所有翻译组成。$G$中的其他一些转换包括 通过垂直直线$L_1$到$P$或平行于$L_1$的任何直线位移为$\overrightarrow{P Q}$的整数倍的直线;通过水直线$L_3=\overleftrightarrow{P Q}$的反射;通过$\pi$围绕$P, Q$的角度旋转，或从$P$平移为$\overrightarrow{P Q}$的整数倍的任何点

$$G=\left\langle s_1, s_2, s_3 \mid s_1^2=s_2^2=s_3^2=1,\left(s_1 s_3\right)^2=1,\left(s_2 s_3\right)^2=1\right\rangle .$$

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