## 数学代写|抽象代数作业代写abstract algebra代考|Properties of Cosets

In this chapter, we will prove the single most important theorem in finite group theory-Lagrange’s Theorem. In his book on abstract algebra, I. N. Herstein likened it to the ABC’s for finite groups. But first we introduce a new and powerful tool for analyzing a group – the notion of a coset. This notion was invented by Galois in 1830, although the term was coined by G. A. Miller in 1910.
Definitions Coset of $H$ in $G$
Let $G$ be a group and let $H$ be a nonempty subset of $G$. For any $a \in G$, the set ${a h \mid h \in H}$ is denoted by $a H$. Analogously, $H a={h a \mid h \in H}$ and $a H a^{-1}=\left{a h a^{-1} \mid h \in\right.$ $H$ }. When $H$ is a subgroup of $G$, the set $a H$ is called the left coset of $H$ in $G$ containing a, whereas $H a$ is called the right coset of $H$ in $G$ containing $a$. In this case, the element $a$ is called the coset representative of $a H$ (or $H a$ ). We use $|a H|$ to denote the number of elements in the set $a H$, and $|H a|$ to denote the number of elements in $H a$.

The three preceding examples illustrate a few facts about cosets that are worthy of our attention. First, cosets are usually not subgroups. Second, $a H$ may be the same as $b H$, even though $a$ is not the same as $b$. Third, since in Example 1 (12) $H=$ ${(12),(132)}$ whereas $H(12)={(12),(123)}, a H$ need not be the same as $\mathrm{Ha}$.

These examples and observations raise many questions. When does $a H=b H$ ? Do $a H$ and $b H$ have any elements in common? When does $a H=H a$ ? Which cosets are subgroups? Why are cosets important? The next lemma and theorem answer these questions. (Analogous results hold for right cosets.)

## 数学代写|抽象代数作业代写abstract algebra代考|An Application of Cosets to Permutation Groups

Lagrange’s Theorem and its corollaries dramatically demonstrate the fruitfulness of the coset concept. We next consider an application of cosets to permutation groups.
Definition Definition Stabilizer of a Point
Let $G$ be a group of permutations of a set $S$. For each $i$ in $S$, let $\operatorname{stab}_G(i)={\phi \in G \mid \phi(i)=i}$. We call $\operatorname{stab}_G(i)$ the stabilizer of $i$ in $G$.

The student should verify that $\operatorname{stab}_G(i)$ is a subgroup of $G$. (See Exercise 43 in Chapter 5.)
Definition Orbit of a Point
Let $G$ be a group of permutations of a set $S$. For each $i$ in $S$, let $\operatorname{orb}_G(i)={\phi(i) \mid \phi \in G}$. The set $\operatorname{orb}_G(i)$ is a subset of $S$ called the orbit of $i$ under $G$. We use $\left|\operatorname{orb}_G(i)\right|$ to denote the number of elements in $\operatorname{orb}_G(i)$.
Example 8 should clarify these two definitions.
EXAMPLE 8 Let $G$ be the following subgroup of $S_8$

Then,
$\operatorname{orb}_G(1)={1,3,2}, \quad \operatorname{stab}_G(1)={(1),(78)}$,
$\operatorname{orb}_G(2)={2,1,3}, \quad \operatorname{stab}_G(2)={(1),(78)}$,
$\operatorname{orb}_G(4)={4,6,5}, \quad \operatorname{stab}_G(4)={(1),(78)}$,
$\operatorname{orb}_G(7)={7,8}, \quad \operatorname{stab}_G(7)={(1),(132)(465),(123)(456)}$.

• EXAMPLE 9 We may view $D_4$ as a group of permutations of a square region. Figure 7.1(a) illustrates the orbit of the point $p$ under $D_4$, and Figure $7.1$ (b) illustrates the orbit of the point $q$ under $D_4$. Observe that $\operatorname{stab}{D_4}(p)=\left{R_0, D\right}$, whereas $\operatorname{stab}{D_4}(q)=\left{R_0\right}$

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## 数学代写|抽象代数作业代写abstract algebra代考|Properties of Cosets

• $H$ 在 $G$
让 $G$ 成为一个群体，让 $H$ 是的非空子集 $G$. 对于任何 $a \in G$ ，集合 $a h \mid h \in H$ 表示为 $a H$. 类似地， , theset —个Hiscalledtheleftcosetof $\mathrm{H}$ in Gcontaininga, whereas哈iscalledtherightcosetof $\mathrm{H} i n \mathrm{G}$ containing—个. Inthiscase, theelement—个iscalledthecosetrepresentativeof-个 (or 哈 ). Weuse|一个 H|todenotethenumberofelementsintheset-个H, and 哈 $\mid$ todenotethenumberofelementsin 哈\$。 前面的三个例子说明了一些值得我们关注的关于陪集的事实。首先，陪集通常不是子群。第二，$a H$可能 与$b H$，虽然$a$不一样$b$. 三、由于在例一 (12)$H=(12),(132)$然而$H(12)=(12),(123), a H$不必相同$\mathrm{Ha}$这些例子和观察提出了许多问题。什么时候$a H=b H$? 做$a H$和$b H$有什么共同点吗? 什么时候$a H=H a$? 哪些陪集是子群? 为什么陪护很重要? 下一个引理和定理回答了这些问题。（类似的结果适用于右陪 集。) ## 数学代写|抽象代数作业代写abstract algebra代考|An Application of Cosets to Permutation Groups 拉格朗日定理及其推论极大地证明了陪集概念的成果。我们接下来考虑将陪集应用于置换群。 定义定义 点的稳定器 稳定器$i$在$G$. 学生应该验证$\operatorname{stab}_G(i)$是一个子群$G$. (参见第 5 章中的练习 43。) 点的定义轨道 Let$G$是一个集合的一组排列$S$. 对于每个在$S$，让$\operatorname{lob}_G(i)=\phi(i) \mid \phi \in G$. 套装$\operatorname{orb}_G(i)$是的一个子集$S$称为轨道$i$在下面$G$. 我们用$\left|\operatorname{orb}_G(i)\right|$来表示元素的数量$\operatorname{orb}_G(i)$. 例 8 应该阐明这两个定义。 例 8 让$G$是以下子群$S_8$然后，$\operatorname{orb}_G(1)=1,3,2, \quad \operatorname{stab}_G(1)=(1),(78)$,$\operatorname{orb}_G(2)=2,1,3, \quad \operatorname{stab}_G(2)=(1),(78)$,$\operatorname{orb}_G(4)=4,6,5, \quad \operatorname{stab}_G(4)=(1),(78)$,$\operatorname{orb}_G(7)=7,8, \quad \operatorname{stab}_G(7)=(1),(132)(465),(123)(456)$. • 例 9 我们可以查看$D_4$作为正方形区域的一组排列。图 7.1(a) 说明了该点的轨道$p$在下面$D_4$，和图7.1(b) 说明该点的轨道$q$在下面$D_4$. 请注意 \operatorname{stab}{D_4}$}$(p)$=\backslash \backslash$left{R_0, D\right}$}\$ ，然而

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